Math 114

Introduction to Linear Algebra

Lecture 1

Math 114Introduction to Linear Algebra Lecture 1 1 / 36

Topics

1 Linear Equations and Linear Systems

2 Linear Combination, Linear Span, Linear Independence
3 Vector and Matrix Arithmetic
4 The Determinant of a Matrix
5 Vector Spaces and Subspaces
6 Basis and Dimension of a Vector Space
7 Linearity and Linear Transformations
8 Eigenvalues and Eigenvectors of a Matrix
9 Similarity and Diagonalizability of Matrices
10 Length, Distance and Orthogonality

Math 114Introduction to Linear Algebra Lecture 1 2 / 36

Why Study Linear Algebra?

Math 114Introduction to Linear Algebra Lecture 1 3 / 36

Why Study Linear Algebra?

Linear Algebra has many applications:

Search Engines
Image Processing
Information Theory
Signal Processing
Control Theory
Networks
etc...

Math 114Introduction to Linear Algebra Lecture 1 3 / 36

Why Study Linear Algebra?

Linear Algebra has many applications:

Search Engines
Image Processing
Information Theory
Signal Processing
Control Theory
Networks
etc...

It is usually the simplest approach to solve more complicated

problems. (Think: curves approximated by tangent lines)

Math 114Introduction to Linear Algebra Lecture 1 3 / 36

Why Study Linear Algebra?

Linear Algebra has many applications:

Search Engines
Image Processing
Information Theory
Signal Processing
Control Theory
Networks
etc...

It is usually the simplest approach to solve more complicated

problems. (Think: curves approximated by tangent lines)
Matrices/vectors are easy to implement in computer algorithms.

Math 114Introduction to Linear Algebra Lecture 1 3 / 36

Objectives

At the end of the semester, I am expecting you to

develop a deeper
understanding of linear
equations/linear systems;
think more critically and use
logical reasoning; and
use precise mathematical
notations and statements.

Math 114Introduction to Linear Algebra Lecture 1 4 / 36

Sets and Set Notation

Math 114Introduction to Linear Algebra Lecture 1 5 / 36

Sets and Set Notation

X = {x1 , ..., xn }

Math 114Introduction to Linear Algebra Lecture 1 5 / 36

Sets and Set Notation

X = {x1 , ..., xn }

N = {0, 1, 2, 3, ...} (natural numbers)

Z = {..., 2, 1, 0, 1, 2, ...} (integers)

Math 114Introduction to Linear Algebra Lecture 1 5 / 36

Sets and Set Notation

X = {x1 , ..., xn }

N = {0, 1, 2, 3, ...} (natural numbers)

Z = {..., 2, 1, 0, 1, 2, ...} (integers)
R = set of all points on a line (real numbers)

Math 114Introduction to Linear Algebra Lecture 1 5 / 36

Sets and Set Notation

X = {x1 , ..., xn }

N = {0, 1, 2, 3, ...} (natural numbers)

Z = {..., 2, 1, 0, 1, 2, ...} (integers)
R = set of all points on a line (real numbers)
set builder notation:

A={ ...
|{z} | ...
|{z} }
|{z}
general description/form such that00 specific characteristics

Math 114Introduction to Linear Algebra Lecture 1 5 / 36

Sets and Set Notation

X = {x1 , ..., xn }

N = {0, 1, 2, 3, ...} (natural numbers)

Z = {..., 2, 1, 0, 1, 2, ...} (integers)
R = set of all points on a line (real numbers)
set builder notation:

A={ ...
|{z} | ...
|{z} }
|{z}
general description/form such that00 specific characteristics

Q = { ba | a, b Z, b 6= 0} (rational numbers)

Math 114Introduction to Linear Algebra Lecture 1 5 / 36

Sets and Set Notation

X = {x1 , ..., xn }

N = {0, 1, 2, 3, ...} (natural numbers)

Z = {..., 2, 1, 0, 1, 2, ...} (integers)
R = set of all points on a line (real numbers)
set builder notation:

A={ ...
|{z} | ...
|{z} }
|{z}
general description/form such that00 specific characteristics

Q = { ba | a, b Z, b 6= 0} (rational numbers)

C = {a + bi | a, b R, i = 1} (complex numbers)

Math 114Introduction to Linear Algebra Lecture 1 5 / 36

Solution Set of Equations

Math 114Introduction to Linear Algebra Lecture 1 6 / 36

Solution Set of Equations

EQUATION

Math 114Introduction to Linear Algebra Lecture 1 6 / 36

Solution Set of Equations
n o n o
EQUATION expression 1 = expression 2

Math 114Introduction to Linear Algebra Lecture 1 6 / 36

Solution Set of Equations
n o n o
EQUATION expression 1 = expression 2

EXPRESSION combo of constants and variables

Math 114Introduction to Linear Algebra Lecture 1 6 / 36

Solution Set of Equations
n o n o
EQUATION expression 1 = expression 2

EXPRESSION combo of constants and variables

3x 2 = 12 x 2 + y 2 = 1 x + 2y 4z = 0

Math 114Introduction to Linear Algebra Lecture 1 6 / 36

Solution Set of Equations
n o n o
EQUATION expression 1 = expression 2

EXPRESSION combo of constants and variables

3x 2 = 12 x 2 + y 2 = 1 x + 2y 4z = 0

SOLUTION

Math 114Introduction to Linear Algebra Lecture 1 6 / 36

Solution Set of Equations
n o n o
EQUATION expression 1 = expression 2

EXPRESSION combo of constants and variables

3x 2 = 12 x 2 + y 2 = 1 x + 2y 4z = 0

SOLUTION value of variables that satisfy equation

Math 114Introduction to Linear Algebra Lecture 1 6 / 36

Solution Set of Equations
n o n o
EQUATION expression 1 = expression 2

EXPRESSION combo of constants and variables

3x 2 = 12 x 2 + y 2 = 1 x + 2y 4z = 0

SOLUTION value of variables that satisfy equation

SOLUTION SET

Math 114Introduction to Linear Algebra Lecture 1 6 / 36

Solution Set of Equations
n o n o
EQUATION expression 1 = expression 2

EXPRESSION combo of constants and variables

3x 2 = 12 x 2 + y 2 = 1 x + 2y 4z = 0

SOLUTION value of variables that satisfy equation

SOLUTION SET collection of all solutions

equation solution set notation

3x 2 = 12

Math 114Introduction to Linear Algebra Lecture 1 6 / 36

Solution Set of Equations
n o n o
EQUATION expression 1 = expression 2

EXPRESSION combo of constants and variables

3x 2 = 12 x 2 + y 2 = 1 x + 2y 4z = 0

SOLUTION value of variables that satisfy equation

SOLUTION SET collection of all solutions

equation solution set notation

3x 2 = 12 x = 2, 2

Math 114Introduction to Linear Algebra Lecture 1 6 / 36

Solution Set of Equations
n o n o
EQUATION expression 1 = expression 2

EXPRESSION combo of constants and variables

3x 2 = 12 x 2 + y 2 = 1 x + 2y 4z = 0

SOLUTION value of variables that satisfy equation

SOLUTION SET collection of all solutions

equation solution set notation

3x 2 = 12 x = 2, 2 SS = {2, 2}

Math 114Introduction to Linear Algebra Lecture 1 6 / 36

Solution Set of Equations
n o n o
EQUATION expression 1 = expression 2

EXPRESSION combo of constants and variables

3x 2 = 12 x 2 + y 2 = 1 x + 2y 4z = 0

SOLUTION value of variables that satisfy equation

SOLUTION SET collection of all solutions

equation solution set notation

3x 2 = 12 x = 2, 2 SS = {2, 2}
x 2 + y 2 = 1

Math 114Introduction to Linear Algebra Lecture 1 6 / 36

Solution Set of Equations
n o n o
EQUATION expression 1 = expression 2

EXPRESSION combo of constants and variables

3x 2 = 12 x 2 + y 2 = 1 x + 2y 4z = 0

SOLUTION value of variables that satisfy equation

SOLUTION SET collection of all solutions

equation solution set notation

3x 2 = 12 x = 2, 2 SS = {2, 2}
x 2 + y 2 = 1 none

Math 114Introduction to Linear Algebra Lecture 1 6 / 36

Solution Set of Equations
n o n o
EQUATION expression 1 = expression 2

EXPRESSION combo of constants and variables

3x 2 = 12 x 2 + y 2 = 1 x + 2y 4z = 0

SOLUTION value of variables that satisfy equation

SOLUTION SET collection of all solutions

equation solution set notation

3x 2 = 12 x = 2, 2 SS = {2, 2}
x 2 + y 2 = 1 none SS = or SS = {}

Math 114Introduction to Linear Algebra Lecture 1 6 / 36

Solution Set of Equations
n o n o
EQUATION expression 1 = expression 2

EXPRESSION combo of constants and variables

3x 2 = 12 x 2 + y 2 = 1 x + 2y 4z = 0

SOLUTION value of variables that satisfy equation

SOLUTION SET collection of all solutions

equation solution set notation

3x 2 = 12 x = 2, 2 SS = {2, 2}
x 2 + y 2 = 1 none SS = or SS = {}
x + 2y 4z = 0

Math 114Introduction to Linear Algebra Lecture 1 6 / 36

Solution Set of Equations
n o n o
EQUATION expression 1 = expression 2

EXPRESSION combo of constants and variables

3x 2 = 12 x 2 + y 2 = 1 x + 2y 4z = 0

SOLUTION value of variables that satisfy equation

SOLUTION SET collection of all solutions

equation solution set notation

3x 2 = 12 x = 2, 2 SS = {2, 2}
x 2 + y 2 = 1 none SS = or SS = {}
x + 2y 4z = 0 infinitely many

Math 114Introduction to Linear Algebra Lecture 1 6 / 36

Solution Set of Equations
n o n o
EQUATION expression 1 = expression 2

EXPRESSION combo of constants and variables

3x 2 = 12 x 2 + y 2 = 1 x + 2y 4z = 0

SOLUTION value of variables that satisfy equation

SOLUTION SET collection of all solutions

equation solution set notation

3x 2 = 12 x = 2, 2 SS = {2, 2}
x 2 + y 2 = 1 none SS = or SS = {}
x + 2y 4z = 0 infinitely many SS = {(x, y , x+2y
4 ) | x, y R}

Math 114Introduction to Linear Algebra Lecture 1 6 / 36

Linear Equations

a1x1 + a2x2 + . . . + an xn = b
(at least one of the ai s is nonzero)

Math 114Introduction to Linear Algebra Lecture 1 7 / 36

Linear Equations

a1x1 + a2x2 + . . . + an xn = b
(at least one of the ai s is nonzero)

example x1 + 2x2 4x3 = 0

Math 114Introduction to Linear Algebra Lecture 1 7 / 36

Linear Equations

a1x1 + a2x2 + . . . + an xn = b
(at least one of the ai s is nonzero)

example x1 + 2x2 4x3 = 0

solution an ordered n-tuple of numbers that satisfy the equation

Math 114Introduction to Linear Algebra Lecture 1 7 / 36

Linear Equations

a1x1 + a2x2 + . . . + an xn = b
(at least one of the ai s is nonzero)

example x1 + 2x2 4x3 = 0

solution an ordered n-tuple of numbers that satisfy the equation
x1
.
(x1 , . . . , xn ) or .
.
xn

Math 114Introduction to Linear Algebra Lecture 1 7 / 36

Linear Equations

a1x1 + a2x2 + . . . + an xn = b
(at least one of the ai s is nonzero)

example x1 + 2x2 4x3 = 0

solution an ordered n-tuple of numbers that satisfy the equation
x1
.
(x1 , . . . , xn ) or .
.
xn
example (2, 1, 1) is a solution of x1 + 2x2 4x3 = 0

Math 114Introduction to Linear Algebra Lecture 1 7 / 36

Linear Equations

a1x1 + a2x2 + . . . + an xn = b
(at least one of the ai s is nonzero)

example x1 + 2x2 4x3 = 0

solution an ordered n-tuple of numbers that satisfy the equation
x1
.
(x1 , . . . , xn ) or .
.
xn
example (2, 1, 1) is a solution of x1 + 2x2 4x3 = 0
(1, 2, 1) is not a solution of x1 + 2x2 4x3 = 0

Math 114Introduction to Linear Algebra Lecture 1 7 / 36

Solution Set of a Linear Equation

# vars equation domain solution set geometry

Math 114Introduction to Linear Algebra Lecture 1 8 / 36

Solution Set of a Linear Equation

# vars equation domain solution set geometry

1 ax = b real line

Math 114Introduction to Linear Algebra Lecture 1 8 / 36

Solution Set of a Linear Equation

# vars equation domain solution set geometry

1 ax = b real line { ba } point

Math 114Introduction to Linear Algebra Lecture 1 8 / 36

Solution Set of a Linear Equation

# vars equation domain solution set geometry

1 ax = b real line { ba } point

2 ax + by = c Cart. plane

Math 114Introduction to Linear Algebra Lecture 1 8 / 36

Solution Set of a Linear Equation

# vars equation domain solution set geometry

1 ax = b real line { ba } point

cax
2 ax + by = c Cart. plane {(x, b
) | x R} line

Math 114Introduction to Linear Algebra Lecture 1 8 / 36

Solution Set of a Linear Equation

# vars equation domain solution set geometry

1 ax = b real line { ba } point

2 ax + by = c Cart. plane {(x, cax

b
) | x R} line
{( cby
a
, y ) | y R}

Math 114Introduction to Linear Algebra Lecture 1 8 / 36

Solution Set of a Linear Equation

# vars equation domain solution set geometry

1 ax = b real line { ba } point

2 ax + by = c Cart. plane {(x, cax

b
) | x R} line
{( cby
a
, y ) | y R}

Euclidean
3 ax + by + cz = d
3D space

Math 114Introduction to Linear Algebra Lecture 1 8 / 36

Solution Set of a Linear Equation

# vars equation domain solution set geometry

1 ax = b real line { ba } point

2 ax + by = c Cart. plane {(x, cax

b
) | x R} line
{( cby
a
, y ) | y R}

Euclidean daxby
3 ax + by + cz = d {(x, y , c
) | x, y R} plane
3D space

Math 114Introduction to Linear Algebra Lecture 1 8 / 36

Solution Set of a Linear Equation

# vars equation domain solution set geometry

1 ax = b real line { ba } point

2 ax + by = c Cart. plane {(x, cax

b
) | x R} line
{( cby
a
, y ) | y R}

Euclidean daxby
3 ax + by + cz = d {(x, y , c
) | x, y R} plane
3D space
daxcz
{(x, b
, z) | x, z R}

Math 114Introduction to Linear Algebra Lecture 1 8 / 36

Solution Set of a Linear Equation

# vars equation domain solution set geometry

1 ax = b real line { ba } point

2 ax + by = c Cart. plane {(x, cax

b
) | x R} line
{( cby
a
, y ) | y R}

Euclidean daxby
3 ax + by + cz = d {(x, y , c
) | x, y R} plane
3D space
daxcz
{(x, b
, z) | x, z R}

{( dbya cz , y , z) | y , z R}

Math 114Introduction to Linear Algebra Lecture 1 8 / 36

Solution Set of a Linear Equation

# vars equation domain solution set geometry

1 ax = b real line { ba } point

2 ax + by = c Cart. plane {(x, cax

b
) | x R} line
{( cby
a
, y ) | y R}

Euclidean daxby
3 ax + by + cz = d {(x, y , c
) | x, y R} plane
3D space
daxcz
{(x, b
, z) | x, z R}

{( dbya cz , y , z) | y , z R}

Euclidean nD-plane
n a1 x1 + +an xn = b
nD-space (hyperplane)

Math 114Introduction to Linear Algebra Lecture 1 8 / 36

Systems of m Linear Equations in n Variables

Math 114Introduction to Linear Algebra Lecture 1 9 / 36

Systems of m Linear Equations in n Variables

a11 x1 + a12 x2 + + a1n xn = b1

a21 x1 + a22 x2 + + a2n xn = b2
.. ..
. .
P
aij xj = bi
.. ..
. .
am1 x1 + am2 x2 + + amn xn = bm

Math 114Introduction to Linear Algebra Lecture 1 9 / 36

Systems of m Linear Equations in n Variables

a11 x1 + a12 x2 + + a1n xn = b1

a21 x1 + a22 x2 + + a2n xn = b2
.. ..
. .
P
aij xj = bi
.. ..
. .
am1 x1 + am2 x2 + + amn xn = bm

aij

Math 114Introduction to Linear Algebra Lecture 1 9 / 36

Systems of m Linear Equations in n Variables

a11 x1 + a12 x2 + + a1n xn = b1

a21 x1 + a22 x2 + + a2n xn = b2
.. ..
. .
P
aij xj = bi
.. ..
. .
am1 x1 + am2 x2 + + amn xn = bm

aij - coefficient of j th variable in i th equation

Math 114Introduction to Linear Algebra Lecture 1 9 / 36

On Solutions of a System

solution of a system - ordered n-tuple of numbers

x1
.
(x1 , . . . , xn ) or ..


xn
that satisfy all m equations

Math 114Introduction to Linear Algebra Lecture 1 10 / 36

On Solutions of a System

solution of a system - ordered n-tuple of numbers

x1
.
(x1 , . . . , xn ) or ..


xn
that satisfy all m equations

solution set of a system - of solution sets of m eqns.

Math 114Introduction to Linear Algebra Lecture 1 10 / 36

On Solutions of a System

solution of a system - ordered n-tuple of numbers

x1
.
(x1 , . . . , xn ) or ..


xn
that satisfy all m equations

solution set of a system - of solution sets of m eqns.

consistent system - has at least one solution

Math 114Introduction to Linear Algebra Lecture 1 10 / 36

On Solutions of a System

solution of a system - ordered n-tuple of numbers

x1
.
(x1 , . . . , xn ) or ..


xn
that satisfy all m equations

solution set of a system - of solution sets of m eqns.

consistent system - has at least one solution

inconsistent system - has no solution

Math 114Introduction to Linear Algebra Lecture 1 10 / 36

Systems of Linear Equations

What are the possible solution sets of the following:

Two Equations, Two Variables

Two Equations, Three Variables

Three Equations, Two Variables

Math 114Introduction to Linear Algebra Lecture 1 11 / 36

Can you figure out the solution set?

x
= 6
S1 = y = 12

z = 5

Math 114Introduction to Linear Algebra Lecture 1 12 / 36

Can you figure out the solution set?

x
= 6 x +y +z
= 1
S1 = y = 12 S2 = x + 2y + 3z = 3

z = 5 x + 4y + 9z = 3

Math 114Introduction to Linear Algebra Lecture 1 12 / 36

Can you figure out the solution set?

x
= 6 x +y +z
= 1
S1 = y = 12 S2 = x + 2y + 3z = 3

z = 5 x + 4y + 9z = 3

x +y +z
= 1
S3 = y + 2z = 2

z = 5

Math 114Introduction to Linear Algebra Lecture 1 12 / 36

Can you figure out the solution set?

x
= 6 x +y +z
= 1
S1 = y = 12 S2 = x + 2y + 3z = 3

z = 5 x + 4y + 9z = 3

x +y +z
= 1
S3 = y + 2z = 2

z = 5

Equivalent systems - systems with the same solution set.

Math 114Introduction to Linear Algebra Lecture 1 12 / 36

Solving Systems

transform system S to system S 0 such that

1 S 0 is equivalent to S

2 S 0 is easy to solve

Math 114Introduction to Linear Algebra Lecture 1 13 / 36

Elementary Operations

Math 114Introduction to Linear Algebra Lecture 1 14 / 36

Elementary Operations

1 Type I (switching equations)

Math 114Introduction to Linear Algebra Lecture 1 14 / 36

Elementary Operations

1 Type I (switching equations)

x +y +z
= 1
Eq2 Eq3
x + 2y + 3z = 3

x + 4y + 9z = 3

Math 114Introduction to Linear Algebra Lecture 1 14 / 36

Elementary Operations

1 Type I (switching equations)

x +y +z
= 1
Eq2 Eq3 x +y +z
= 1
x + 2y + 3z = 3 x + 4y + 9z = 3

x + 4y + 9z = 3 x + 2y + 3z = 3

Math 114Introduction to Linear Algebra Lecture 1 14 / 36

Elementary Operations

1 Type I (switching equations)

2 Type II (scaling an equation by a nonzero number)

Math 114Introduction to Linear Algebra Lecture 1 14 / 36

Elementary Operations

1 Type I (switching equations)

2 Type II (scaling an equation by a nonzero number)

x +y +z
= 1
Eq1 2Eq1
x + 2y + 3z = 3

x + 4y + 9z = 3

Math 114Introduction to Linear Algebra Lecture 1 14 / 36

Elementary Operations

1 Type I (switching equations)

2 Type II (scaling an equation by a nonzero number)

x +y +z
= 1
Eq1 2Eq1 2x + 2y + 2z
= 2
x + 2y + 3z = 3 x + 2y + 3z = 3

x + 4y + 9z = 3 x + 4y + 9z = 3

Math 114Introduction to Linear Algebra Lecture 1 14 / 36

Elementary Operations

1 Type I (switching equations)

2 Type II (scaling an equation by a nonzero number)
3 Type III (replace an eqn with the sum of itself and a multiple of
another eqn)

Math 114Introduction to Linear Algebra Lecture 1 14 / 36

Elementary Operations

1 Type I (switching equations)

2 Type II (scaling an equation by a nonzero number)
3 Type III (replace an eqn with the sum of itself and a multiple of
another eqn)

x +y +z
= 1
Eq3 Eq3 4Eq1
x + 2y + 3z = 3

x + 4y + 9z = 3

Math 114Introduction to Linear Algebra Lecture 1 14 / 36

Elementary Operations

1 Type I (switching equations)

2 Type II (scaling an equation by a nonzero number)
3 Type III (replace an eqn with the sum of itself and a multiple of
another eqn)

x +y +z = 1

Eq3 Eq3 4Eq1 x + y + z = 1

x + 2y + 3z = 3
x + 2y + 3z = 3
x + 4y + 9z = 3

3x + 5z = 7

Math 114Introduction to Linear Algebra Lecture 1 14 / 36

Elementary Operations

1 Type I (switching equations)

2 Type II (scaling an equation by a nonzero number)
3 Type III (replace an eqn with the sum of itself and a multiple of
another eqn)

Theorem
Two systems have the same solution set if and only if one can be obtained
from the other via elementary operations.

Math 114Introduction to Linear Algebra Lecture 1 14 / 36

Gauss-Jordan Elimination

x +y +z
= 1
x + 2y + 3z = 3

x + 4y + 9z = 3

Math 114Introduction to Linear Algebra Lecture 1 15 / 36

Gauss-Jordan Elimination

x +y +z
= 1 E2 E2 E1
x + 2y + 3z = 3

x + 4y + 9z = 3

Math 114Introduction to Linear Algebra Lecture 1 15 / 36

Gauss-Jordan Elimination

x +y +z
= 1 E2 E2 E1 x +y +z
= 1
x + 2y + 3z = 3 y + 2z = 2

x + 4y + 9z = 3 x + 4y + 9z = 3

Math 114Introduction to Linear Algebra Lecture 1 15 / 36

Gauss-Jordan Elimination

x +y +z
= 1 E2 E2 E1 x +y +z
= 1
x + 2y + 3z = 3 y + 2z = 2

x + 4y + 9z = 3 x + 4y + 9z = 3

E3 E3 E1

Math 114Introduction to Linear Algebra Lecture 1 15 / 36

Gauss-Jordan Elimination

x +y +z
= 1 E2 E2 E1 x +y +z
= 1
x + 2y + 3z = 3 y + 2z = 2

x + 4y + 9z = 3 x + 4y + 9z = 3


E3 E3 E1 x +y +z
= 1
y + 2z = 2

3y + 8z = 4

Math 114Introduction to Linear Algebra Lecture 1 15 / 36

Gauss-Jordan Elimination

x +y +z
= 1 E2 E2 E1 x +y +z
= 1
x + 2y + 3z = 3 y + 2z = 2

x + 4y + 9z = 3 x + 4y + 9z = 3


E3 E3 E1 x +y +z
= 1
y + 2z = 2

3y + 8z = 4

E3 E3 3E2

Math 114Introduction to Linear Algebra Lecture 1 15 / 36

Gauss-Jordan Elimination

x +y +z
= 1 E2 E2 E1 x +y +z
= 1
x + 2y + 3z = 3 y + 2z = 2

x + 4y + 9z = 3 x + 4y + 9z = 3


E3 E3 E1 x +y +z
= 1
y + 2z = 2

3y + 8z = 4


E3 E3 3E2 x +y +z
= 1
y + 2z = 2

2z = 10

Math 114Introduction to Linear Algebra Lecture 1 15 / 36

Gauss-Jordan Elimination

x +y +z
= 1 E2 E2 E1 x +y +z
= 1
x + 2y + 3z = 3 y + 2z = 2

x + 4y + 9z = 3 x + 4y + 9z = 3


E3 E3 E1 x +y +z
= 1
y + 2z = 2

3y + 8z = 4


E3 E3 3E2 x +y +z
= 1
y + 2z = 2

2z = 10

E3 12 E3

Math 114Introduction to Linear Algebra Lecture 1 15 / 36

Gauss-Jordan Elimination

x +y +z
= 1 E2 E2 E1 x +y +z
= 1
x + 2y + 3z = 3 y + 2z = 2

x + 4y + 9z = 3 x + 4y + 9z = 3


E3 E3 E1 x +y +z
= 1
y + 2z = 2

3y + 8z = 4


E3 E3 3E2 x +y +z
= 1
y + 2z = 2

2z = 10


E3 12 E3 x +y +z
= 1
y + 2z = 2

z = 5

Math 114Introduction to Linear Algebra Lecture 1 15 / 36

Back-Substitution

x +y +z
= 1
y + 2z = 2

z = 5

Math 114Introduction to Linear Algebra Lecture 1 16 / 36

Back-Substitution

x +y +z
= 1
E2 E2 2E3
y + 2z = 2

z = 5

Math 114Introduction to Linear Algebra Lecture 1 16 / 36

Back-Substitution

x +y +z
= 1
E2 E2 2E3 x +y +z
= 1
y + 2z = 2 y = 12

z = 5 z = 5

Math 114Introduction to Linear Algebra Lecture 1 16 / 36

Back-Substitution

x +y +z
= 1
E2 E2 2E3 x +y +z
= 1
y + 2z = 2 y = 12

z = 5 z = 5

E1 E1 E2

Math 114Introduction to Linear Algebra Lecture 1 16 / 36

Back-Substitution

x +y +z
= 1
E2 E2 2E3 x +y +z
= 1
y + 2z = 2 y = 12

z = 5 z = 5


x
+z = 11
E1 E1 E2
y = 12

z = 5

Math 114Introduction to Linear Algebra Lecture 1 16 / 36

Back-Substitution

x +y +z
= 1
E2 E2 2E3 x +y +z
= 1
y + 2z = 2 y = 12

z = 5 z = 5


x
+z = 11
E1 E1 E2
y = 12

z = 5

E1 E1 E3

Math 114Introduction to Linear Algebra Lecture 1 16 / 36

Back-Substitution

x +y +z
= 1
E2 E2 2E3 x +y +z
= 1
y + 2z = 2 y = 12

z = 5 z = 5


x
+z = 11
E1 E1 E2
y = 12

z = 5


x
= 6
E1 E1 E3
y = 12

z = 5

Math 114Introduction to Linear Algebra Lecture 1 16 / 36

Back-Substitution

x +y +z
= 1
E2 E2 2E3 x +y +z
= 1
y + 2z = 2 y = 12

z = 5 z = 5


x
+z = 11
E1 E1 E2
y = 12

z = 5


x
= 6
E1 E1 E3
y = 12

z = 5

SS = {(6, 12, 5)}

Math 114Introduction to Linear Algebra Lecture 1 16 / 36
Coefficient Matrix of a System

a11 x1 + a12 x2 + + a1n xn = b1 a11 a12 a1n

a21 x1 + a22 x2 + + a2n xn = b2 a21 a12 a2n
.. .. .. .. ..

..
. . . . . .
P
aij xj = bi a
i1 ai2 ain
.. .. .
.. .. .. ..
. . . . .

am1 x1 + am2 x2 + + amn xn = bm am1 am2 amn

Math 114Introduction to Linear Algebra Lecture 1 17 / 36

Coefficient Matrix of a System

a11 x1 + a12 x2 + + a1n xn = b1 a11 a12 a1n

a21 x1 + a22 x2 + + a2n xn = b2 a21 a12 a2n
.. .. .. .. ..

..
. . . . . .
P
aij xj = bi a
i1 ai2 ain
.. .. .
.. .. .. ..
. . . . .

am1 x1 + am2 x2 + + amn xn = bm am1 am2 amn

an m n matrix/array

Math 114Introduction to Linear Algebra Lecture 1 17 / 36

Coefficient Matrix of a System

a11 x1 + a12 x2 + + a1n xn = b1 a11 a12 a1n

a21 x1 + a22 x2 + + a2n xn = b2 a21 a12 a2n
.. .. .. .. ..

..
. . . . . .
P
aij xj = bi a
i1 ai2 ain
.. .. .
.. .. .. ..
. . . . .

am1 x1 + am2 x2 + + amn xn = bm am1 am2 amn

an m n matrix/array
rows - represent m equations
columns - represent n variables

Math 114Introduction to Linear Algebra Lecture 1 17 / 36

Augmented Matrix of a System

a11 x1 + a12 x2 + + a1n xn = b1 a11 a12 a1n b1

a21 x1 + a22 x2 + + a2n xn = b2 a21 a22 a2n b2
.. .. .. .. .. ..

..
. . . . . . .
P
aij xj = bi a
i1 ai2 ain bi
.. .. .
.. .. .. .. ..
. . . . . .

am1 x1 + am2 x2 + + amn xn = bm am1 am2 amn bm

Math 114Introduction to Linear Algebra Lecture 1 18 / 36

Augmented Matrix of a System

a11 x1 + a12 x2 + + a1n xn = b1 a11 a12 a1n b1

a21 x1 + a22 x2 + + a2n xn = b2 a21 a22 a2n b2
.. .. .. .. .. ..

..
. . . . . . .
P
aij xj = bi a
i1 ai2 ain bi
.. .. .
.. .. .. .. ..
. . . . . .

am1 x1 + am2 x2 + + amn xn = bm am1 am2 amn bm

an m (n + 1) matrix

Math 114Introduction to Linear Algebra Lecture 1 18 / 36

Examples: Coefficient Matrix and Augmented Matrix

x
= 6
S1 = y = 12

z = 5

Math 114Introduction to Linear Algebra Lecture 1 19 / 36

Examples: Coefficient Matrix and Augmented Matrix

x
= 6 1 0 0
S1 = y = 12 0 1 0


z = 5 0 0 1

Math 114Introduction to Linear Algebra Lecture 1 19 / 36

Examples: Coefficient Matrix and Augmented Matrix

x
= 6 1 0 0 1 0 0 6
S1 = y = 12 0 1 0 0 1 0 12


z = 5 0 0 1 0 0 1 5

Math 114Introduction to Linear Algebra Lecture 1 19 / 36

Examples: Coefficient Matrix and Augmented Matrix

x
= 6 1 0 0 1 0 0 6
S1 = y = 12 0 1 0 0 1 0 12


z = 5 0 0 1 0 0 1 5

x +y +z
= 1
S2 = x + 2y + 3z = 3

x + 4y + 9z = 3

Math 114Introduction to Linear Algebra Lecture 1 19 / 36

Examples: Coefficient Matrix and Augmented Matrix

x
= 6 1 0 0 1 0 0 6
S1 = y = 12 0 1 0 0 1 0 12


z = 5 0 0 1 0 0 1 5

x +y +z
= 1 1 1 1
S2 = x + 2y + 3z = 3 1 2 3


x + 4y + 9z = 3 1 4 9

Math 114Introduction to Linear Algebra Lecture 1 19 / 36

Examples: Coefficient Matrix and Augmented Matrix

x
= 6 1 0 0 1 0 0 6
S1 = y = 12 0 1 0 0 1 0 12


z = 5 0 0 1 0 0 1 5

x +y +z
= 1 1 1 1 1 1 1 1
S2 = x + 2y + 3z = 3 1 2 3 1 2 3 3


x + 4y + 9z = 3 1 4 9 1 4 9 3

Math 114Introduction to Linear Algebra Lecture 1 19 / 36

Examples: Coefficient Matrix and Augmented Matrix

x
= 6 1 0 0 1 0 0 6
S1 = y = 12 0 1 0 0 1 0 12


z = 5 0 0 1 0 0 1 5

x +y +z
= 1 1 1 1 1 1 1 1
S2 = x + 2y + 3z = 3 1 2 3 1 2 3 3


x + 4y + 9z = 3 1 4 9 1 4 9 3

x +y +z
= 1
S3 = y + 2z = 2

z = 5

Math 114Introduction to Linear Algebra Lecture 1 19 / 36

Examples: Coefficient Matrix and Augmented Matrix

x
= 6 1 0 0 1 0 0 6
S1 = y = 12 0 1 0 0 1 0 12


z = 5 0 0 1 0 0 1 5

x +y +z
= 1 1 1 1 1 1 1 1
S2 = x + 2y + 3z = 3 1 2 3 1 2 3 3


x + 4y + 9z = 3 1 4 9 1 4 9 3

x +y +z
= 1 1 1 1 1 1 1 1
S3 = y + 2z = 2 0 1 2 0 1 2 2


z = 5 0 0 1 0 0 1 5

Math 114Introduction to Linear Algebra Lecture 1 19 / 36

Elementary Row Operations on Matrices

Math 114Introduction to Linear Algebra Lecture 1 20 / 36

Elementary Row Operations on Matrices

1 Type I (switching rows)

Math 114Introduction to Linear Algebra Lecture 1 20 / 36

Elementary Row Operations on Matrices

1 Type I (switching rows)

1 1 1 1
R2 R3
1 2 3 3

1 4 9 3

Math 114Introduction to Linear Algebra Lecture 1 20 / 36

Elementary Row Operations on Matrices

1 Type I (switching rows)

1 1 1 1 1 1 1 1
R2 R3
1 2 3 3 1 4 9 3

1 4 9 3 1 2 3 3

Math 114Introduction to Linear Algebra Lecture 1 20 / 36

Elementary Row Operations on Matrices

1 Type I (switching rows)

2 Type II (scaling a row by a nonzero number)

Math 114Introduction to Linear Algebra Lecture 1 20 / 36

Elementary Row Operations on Matrices

1 Type I (switching rows)

2 Type II (scaling a row by a nonzero number)

1 1 1 1
R1 2R1
1 2 3 3

1 4 9 3

Math 114Introduction to Linear Algebra Lecture 1 20 / 36

Elementary Row Operations on Matrices

1 Type I (switching rows)

2 Type II (scaling a row by a nonzero number)

1 1 1 1 2 2 2 2
R1 2R1
1 2 3 3 1 2 3 3

1 4 9 3 1 4 9 3

Math 114Introduction to Linear Algebra Lecture 1 20 / 36

Elementary Row Operations on Matrices

1 Type I (switching rows)

2 Type II (scaling a row by a nonzero number)
3 Type III (replace a row with the sum of itself and a multiple of
another row)

Math 114Introduction to Linear Algebra Lecture 1 20 / 36

Elementary Row Operations on Matrices

1 Type I (switching rows)

2 Type II (scaling a row by a nonzero number)
3 Type III (replace a row with the sum of itself and a multiple of
another row)

1 1 1 1
R3 R3 4R1
1 2 3 3

1 4 9 3

Math 114Introduction to Linear Algebra Lecture 1 20 / 36

Elementary Row Operations on Matrices

1 Type I (switching rows)

2 Type II (scaling a row by a nonzero number)
3 Type III (replace a row with the sum of itself and a multiple of
another row)

1 1 1 1 1 1 1 1
R3 R3 4R1
1 2 3 3 1 2 3 3

1 4 9 3 3 0 5 7

Math 114Introduction to Linear Algebra Lecture 1 20 / 36

Row Equivalence of Matrices

Two matrices are row equivalent if one can be obtained from the other
via elementary row operations.
Notation: A B means A and B are row equivalent matrices.

Math 114Introduction to Linear Algebra Lecture 1 21 / 36

Row Equivalence of Matrices

Two matrices are row equivalent if one can be obtained from the other
via elementary row operations.
Notation: A B means A and B are row equivalent matrices.
Theorem
If two systems are equivalent, then their corresponding augmented
matrices are row equivalent.

Math 114Introduction to Linear Algebra Lecture 1 21 / 36

Echelon Forms

Math 114Introduction to Linear Algebra Lecture 1 22 / 36

Echelon Forms

To solve a system
1 find its augmented matrix
2 apply row operations to reduce to echelon form

3 describe solution set using echelon form

Math 114Introduction to Linear Algebra Lecture 1 22 / 36

RRF and RREF

A matrix is in ...
Row Reduced Form (RRF) if

Math 114Introduction to Linear Algebra Lecture 1 23 / 36

RRF and RREF

A matrix is in ...
Row Reduced Form (RRF) if
1 All zero rows are at the bottom.

Math 114Introduction to Linear Algebra Lecture 1 23 / 36

RRF and RREF

A matrix is in ...
Row Reduced Form (RRF) if
1 All zero rows are at the bottom.
2 The *leading entry of a nonzero row is strictly on the left of the
leading entry of any nonzero row below it.

Math 114Introduction to Linear Algebra Lecture 1 23 / 36

RRF and RREF

A matrix is in ...
Row Reduced Form (RRF) if
1 All zero rows are at the bottom.
2 The *leading entry of a nonzero row is strictly on the left of the
leading entry of any nonzero row below it.

*leading entry of a nonzero row - first nonzero entry from the left

Row Reduced Echelon Form (RREF) if

Math 114Introduction to Linear Algebra Lecture 1 23 / 36

RRF and RREF

A matrix is in ...
Row Reduced Form (RRF) if
1 All zero rows are at the bottom.
2 The *leading entry of a nonzero row is strictly on the left of the
leading entry of any nonzero row below it.

*leading entry of a nonzero row - first nonzero entry from the left

Row Reduced Echelon Form (RREF) if

1 The matrix is in RRF.

Math 114Introduction to Linear Algebra Lecture 1 23 / 36

RRF and RREF

A matrix is in ...
Row Reduced Form (RRF) if
1 All zero rows are at the bottom.
2 The *leading entry of a nonzero row is strictly on the left of the
leading entry of any nonzero row below it.

*leading entry of a nonzero row - first nonzero entry from the left

Row Reduced Echelon Form (RREF) if

1 The matrix is in RRF.
2 The leading entry in each nonzero row is 1.

Math 114Introduction to Linear Algebra Lecture 1 23 / 36

RRF and RREF

A matrix is in ...
Row Reduced Form (RRF) if
1 All zero rows are at the bottom.
2 The *leading entry of a nonzero row is strictly on the left of the
leading entry of any nonzero row below it.

*leading entry of a nonzero row - first nonzero entry from the left

Row Reduced Echelon Form (RREF) if

1 The matrix is in RRF.
2 The leading entry in each nonzero row is 1.
3 Each leading 1 is the only nonzero entry in its column.

Math 114Introduction to Linear Algebra Lecture 1 23 / 36


2 1 0 0

0 1 0 1

0 0 1 2

0 0 2 4

RRF?

Math 114Introduction to Linear Algebra Lecture 1 24 / 36


2 1 0 0

0 1 0 1

0 0 1 2

0 0 2 4

RRF? No.

Math 114Introduction to Linear Algebra Lecture 1 24 / 36


2 1 0 0

0 1 0 1

0 0 1 2

0 0 2 4

RRF? No.

RREF?

Math 114Introduction to Linear Algebra Lecture 1 24 / 36


2 1 0 0

0 1 0 1

0 0 1 2

0 0 2 4

RRF? No.

RREF? No.

Math 114Introduction to Linear Algebra Lecture 1 24 / 36


2 1 0 1 0
0 0 1 0 2

0 1 0 1 1

RRF?

Math 114Introduction to Linear Algebra Lecture 1 25 / 36


2 1 0 1 0
0 0 1 0 2

0 1 0 1 1

RRF? No.

Math 114Introduction to Linear Algebra Lecture 1 25 / 36


2 1 0 1 0
0 0 1 0 2

0 1 0 1 1

RRF? No.

RREF?

Math 114Introduction to Linear Algebra Lecture 1 25 / 36


2 1 0 1 0
0 0 1 0 2

0 1 0 1 1

RRF? No.

RREF? No.

Math 114Introduction to Linear Algebra Lecture 1 25 / 36


2 1 0 0
0 0 1 2

0 0 0 1

RRF?

Math 114Introduction to Linear Algebra Lecture 1 26 / 36


2 1 0 0
0 0 1 2

0 0 0 1

RRF? Yes.

Math 114Introduction to Linear Algebra Lecture 1 26 / 36


2 1 0 0
0 0 1 2

0 0 0 1

RRF? Yes.

RREF?

Math 114Introduction to Linear Algebra Lecture 1 26 / 36


2 1 0 0
0 0 1 2

0 0 0 1

RRF? Yes.

RREF? No.

Math 114Introduction to Linear Algebra Lecture 1 26 / 36


2 1 0 0
0 0 1 2

0 0 0 1

RRF? Yes.

RREF? No.

pivot columns?

Math 114Introduction to Linear Algebra Lecture 1 26 / 36


2 1 0 0
0 0 1 2

0 0 0 1

RRF? Yes.

RREF? No.

pivot columns? 1st and 3rd

Math 114Introduction to Linear Algebra Lecture 1 26 / 36


2 0 0 3

0 1 0 1

0 0 1 2

0 0 0 0

RRF?

Math 114Introduction to Linear Algebra Lecture 1 27 / 36


2 0 0 3

0 1 0 1

0 0 1 2

0 0 0 0

RRF? Yes.

Math 114Introduction to Linear Algebra Lecture 1 27 / 36


2 0 0 3

0 1 0 1

0 0 1 2

0 0 0 0

RRF? Yes.

RREF?

Math 114Introduction to Linear Algebra Lecture 1 27 / 36


2 0 0 3

0 1 0 1

0 0 1 2

0 0 0 0

RRF? Yes.

RREF? No.

Math 114Introduction to Linear Algebra Lecture 1 27 / 36


2 0 0 3

0 1 0 1

0 0 1 2

0 0 0 0

RRF? Yes.

RREF? No.

pivot columns?

Math 114Introduction to Linear Algebra Lecture 1 27 / 36


2 0 0 3

0 1 0 1

0 0 1 2

0 0 0 0

RRF? Yes.

RREF? No.

pivot columns? 1st,2nd and 3rd

Math 114Introduction to Linear Algebra Lecture 1 27 / 36


1 1 0 0
0 0 1 2

0 0 0 1

RRF?

Math 114Introduction to Linear Algebra Lecture 1 28 / 36


1 1 0 0
0 0 1 2

0 0 0 1

RRF? Yes.

Math 114Introduction to Linear Algebra Lecture 1 28 / 36


1 1 0 0
0 0 1 2

0 0 0 1

RRF? Yes.

RREF?

Math 114Introduction to Linear Algebra Lecture 1 28 / 36


1 1 0 0
0 0 1 2

0 0 0 1

RRF? Yes.

RREF? No.

Math 114Introduction to Linear Algebra Lecture 1 28 / 36


1 1 0 0
0 0 1 2

0 0 0 1

RRF? Yes.

RREF? No.

pivot columns?

Math 114Introduction to Linear Algebra Lecture 1 28 / 36


1 1 0 0
0 0 1 2

0 0 0 1

RRF? Yes.

RREF? No.

pivot columns? 1st, 3rd and 4th

Math 114Introduction to Linear Algebra Lecture 1 28 / 36


1 0 0 1 0
0 1 0 1 1

0 0 0 0 0

RRF?

Math 114Introduction to Linear Algebra Lecture 1 29 / 36


1 0 0 1 0
0 1 0 1 1

0 0 0 0 0

RRF? Yes.

Math 114Introduction to Linear Algebra Lecture 1 29 / 36


1 0 0 1 0
0 1 0 1 1

0 0 0 0 0

RRF? Yes.

RREF?

Math 114Introduction to Linear Algebra Lecture 1 29 / 36


1 0 0 1 0
0 1 0 1 1

0 0 0 0 0

RRF? Yes.

RREF? Yes.

Math 114Introduction to Linear Algebra Lecture 1 29 / 36


1 0 0 1 0
0 1 0 1 1

0 0 0 0 0

RRF? Yes.

RREF? Yes.

pivot columns?

Math 114Introduction to Linear Algebra Lecture 1 29 / 36


1 0 0 1 0
0 1 0 1 1

0 0 0 0 0

RRF? Yes.

RREF? Yes.

pivot columns? 1st and 2nd

Math 114Introduction to Linear Algebra Lecture 1 29 / 36

Row Reduction

Not all matrices are in RRF/RREF...

Math 114Introduction to Linear Algebra Lecture 1 30 / 36

Row Reduction

Not all matrices are in RRF/RREF... BUT...

Math 114Introduction to Linear Algebra Lecture 1 30 / 36

Row Reduction

Not all matrices are in RRF/RREF... BUT...

Theorem
Any A is row equivalent to a matrix B that is in RRF. (B is not unique).

Math 114Introduction to Linear Algebra Lecture 1 30 / 36

Row Reduction

Not all matrices are in RRF/RREF... BUT...

Theorem
Any A is row equivalent to a matrix B that is in RRF. (B is not unique).

Theorem
Any A is row equivalent to a matrix B that is in in RREF. (B is unique).

Math 114Introduction to Linear Algebra Lecture 1 30 / 36

Row Reduction

Not all matrices are in RRF/RREF... BUT...

Theorem
Any A is row equivalent to a matrix B that is in RRF. (B is not unique).

Theorem
Any A is row equivalent to a matrix B that is in in RREF. (B is unique).

Math 114Introduction to Linear Algebra Lecture 1 30 / 36

Row Reduction

Not all matrices are in RRF/RREF... BUT...

Theorem
Any A is row equivalent to a matrix B that is in RRF. (B is not unique).

Theorem
Any A is row equivalent to a matrix B that is in in RREF. (B is unique).

Row Reduction - process of finding RRF or RREF of a matrix

Math 114Introduction to Linear Algebra Lecture 1 30 / 36

Row Reduction

Not all matrices are in RRF/RREF... BUT...

Theorem
Any A is row equivalent to a matrix B that is in RRF. (B is not unique).

Theorem
Any A is row equivalent to a matrix B that is in in RREF. (B is unique).

Row Reduction - process of finding RRF or RREF of a matrix

Apply Gauss-Jordan elimination idea to get RRF.

Math 114Introduction to Linear Algebra Lecture 1 30 / 36

Row Reduction

Not all matrices are in RRF/RREF... BUT...

Theorem
Any A is row equivalent to a matrix B that is in RRF. (B is not unique).

Theorem
Any A is row equivalent to a matrix B that is in in RREF. (B is unique).

Row Reduction - process of finding RRF or RREF of a matrix

Apply Gauss-Jordan elimination idea to get RRF.
Apply Gauss-Jordan elimination and back-substitution idea to get
RREF.

Math 114Introduction to Linear Algebra Lecture 1 30 / 36

Row Reduction

Not all matrices are in RRF/RREF... BUT...

Theorem
Any A is row equivalent to a matrix B that is in RRF. (B is not unique).

Theorem
Any A is row equivalent to a matrix B that is in in RREF. (B is unique).

Row Reduction - process of finding RRF or RREF of a matrix

Apply Gauss-Jordan elimination idea to get RRF.
Apply Gauss-Jordan elimination and back-substitution idea to get
RREF.

Math 114Introduction to Linear Algebra Lecture 1 30 / 36

Two Fundamental Questions

Given a system of linear equations S,

1 Does S have any solution?
2 If it exists, is the solution unique?

Math 114Introduction to Linear Algebra Lecture 1 31 / 36

Existence of Solution

To determine consistency/inconsistency of a linear system, row reduce to

RRF.

Math 114Introduction to Linear Algebra Lecture 1 32 / 36

Existence of Solution

To determine consistency/inconsistency of a linear system, row reduce to

RRF.

x1 + x2 +x3 = 5

x1 x2 +x3 = 1

4x2 = 0

(
x1 2x2 = 1
x1 + 3x2 = 3

Math 114Introduction to Linear Algebra Lecture 1 32 / 36

Existence of Solution

To determine consistency/inconsistency of a linear system, row reduce to

RRF.

x1 + x2 +x3 = 5

x1 x2 +x3 = 1

4x2 = 0

(
x1 2x2 = 1
x1 + 3x2 = 3

Rule
If there is a pivot in the last column, then the system is inconsistent (no
solution). Otherwise, it is consistent.

Math 114Introduction to Linear Algebra Lecture 1 32 / 36

Uniqueness of Solutions

To determine the solution set of a linear system, row reduce to RREF.

Math 114Introduction to Linear Algebra Lecture 1 33 / 36

Uniqueness of Solutions

To determine the solution set of a linear system, row reduce to RREF.

Rule
If all columns, except the last, have a pivot, then the solution is unique.
Columns without a pivot correspond to free variables.
(
x1 2x2 = 1
1
x1 + 3x2 = 3
(
x1 + x2 + x3 = 6
2
2x1 + 2x2 + x3 = 8

Math 114Introduction to Linear Algebra Lecture 1 33 / 36

Exercises

Given the row reduced augmented matrix of the system, determine how
many solutions there are.

2 1 0 0

0 1 0 1
(a)

0 0 1 2

0 0 0 0

Math 114Introduction to Linear Algebra Lecture 1 34 / 36

Exercises

Given the row reduced augmented matrix of the system, determine how
many solutions there are.

2 1 0 0

0 1 0 1
(a)
one
0 0 1 2

0 0 0 0

Math 114Introduction to Linear Algebra Lecture 1 34 / 36

Exercises

Given the row reduced augmented matrix of the system, determine how
many solutions there are.

2 1 0 0

0 1 0 1
(a)
one
0 0 1 2

0 0 0 0

2 1 0 0
(b) 0 0 1 2

0 0 0 1

Math 114Introduction to Linear Algebra Lecture 1 34 / 36

Exercises

Given the row reduced augmented matrix of the system, determine how
many solutions there are.

2 1 0 0

0 1 0 1
(a)
one
0 0 1 2

0 0 0 0

2 1 0 0
(b) 0 0 1 2 none

0 0 0 1

Math 114Introduction to Linear Algebra Lecture 1 34 / 36

Exercises

Given the row reduced augmented matrix of the system, determine how
many solutions there are.

2 1 0 0

0 1 0 1
(a)
one
0 0 1 2

0 0 0 0

2 1 0 0
(b) 0 0 1 2 none

0 0 0 1
" #
2 1 0 1 0
(c)
0 1 0 1 1

Math 114Introduction to Linear Algebra Lecture 1 34 / 36

Exercises

Given the row reduced augmented matrix of the system, determine how
many solutions there are.

2 1 0 0

0 1 0 1
(a)
one
0 0 1 2

0 0 0 0

2 1 0 0
(b) 0 0 1 2 none

0 0 0 1
" #
2 1 0 1 0
(c) infinitely many
0 1 0 1 1

Math 114Introduction to Linear Algebra Lecture 1 34 / 36

Given the RREF of the augmented matrix of the system, write the solution
set explicitly (using free variables, if any).

1 0 0 0

0 1 0 1
(a)
0 0 1

2

0 0 0 0

Math 114Introduction to Linear Algebra Lecture 1 35 / 36

Given the RREF of the augmented matrix of the system, write the solution
set explicitly (using free variables, if any).

1 0 0 0

0 1 0 1
(a)
0 0 1
SS = {(0, 1, 2)}
2

0 0 0 0

Math 114Introduction to Linear Algebra Lecture 1 35 / 36

Given the RREF of the augmented matrix of the system, write the solution
set explicitly (using free variables, if any).

1 0 0 0

0 1 0 1
(a)
0 0 1
SS = {(0, 1, 2)}
2

0 0 0 0

1 0 0 6
(b) 0 1 0 3

0 0 1 4

Math 114Introduction to Linear Algebra Lecture 1 35 / 36

Given the RREF of the augmented matrix of the system, write the solution
set explicitly (using free variables, if any).

1 0 0 0

0 1 0 1
(a)
0 0 1
SS = {(0, 1, 2)}
2

0 0 0 0

1 0 0 6
(b) 0 1 0 3 SS = {(6, 3, 4)}

0 0 1 4

Math 114Introduction to Linear Algebra Lecture 1 35 / 36

Given the RREF of the augmented matrix of the system, write the solution
set explicitly (using free variables, if any).

1 0 0 0

0 1 0 1
(a)
0 0 1
SS = {(0, 1, 2)}
2

0 0 0 0

1 0 0 6
(b) 0 1 0 3 SS = {(6, 3, 4)}

0 0 1 4

1 0 3 4
(c) 0 1 0 2

0 0 0 1

Math 114Introduction to Linear Algebra Lecture 1 35 / 36

Given the RREF of the augmented matrix of the system, write the solution
set explicitly (using free variables, if any).

1 0 0 0

0 1 0 1
(a)
0 0 1
SS = {(0, 1, 2)}
2

0 0 0 0

1 0 0 6
(b) 0 1 0 3 SS = {(6, 3, 4)}

0 0 1 4

1 0 3 4
(c) 0 1 0 2 SS =

0 0 0 1

Math 114Introduction to Linear Algebra Lecture 1 35 / 36

Parametric Solution set

Given the RREF of the augmented matrix of the system, write the solution
set explicitly (using free variables, if any).
" #
1 0 0 1 0
1
0 1 1 3 1

Math 114Introduction to Linear Algebra Lecture 1 36 / 36

Parametric Solution set

Given the RREF of the augmented matrix of the system, write the solution
set explicitly (using free variables, if any).
" #
1 0 0 1 0
1
0 1 1 3 1
SS = {( , , x3 , x4 ) | x3 , x4 R}

Math 114Introduction to Linear Algebra Lecture 1 36 / 36

Parametric Solution set

Given the RREF of the augmented matrix of the system, write the solution
set explicitly (using free variables, if any).
" #
1 0 0 1 0
1
0 1 1 3 1
SS = {( x4 , , x3 , x4 ) | x3 , x4 R}

Math 114Introduction to Linear Algebra Lecture 1 36 / 36

Parametric Solution set

Given the RREF of the augmented matrix of the system, write the solution
set explicitly (using free variables, if any).
" #
1 0 0 1 0
1
0 1 1 3 1
SS = {( x4 , 1 + x3 3x4 , x3 , x4 ) | x3 , x4 R}

Math 114Introduction to Linear Algebra Lecture 1 36 / 36

Parametric Solution set

Given the RREF of the augmented matrix of the system, write the solution
set explicitly (using free variables, if any).
" #
1 0 0 1 0
1
0 1 1 3 1
SS = {( x4 , 1 + x3 3x4 , x3 , x4 ) | x3 , x4 R}
" #
1 0 0 0 4
2
0 1 0 0 11

Math 114Introduction to Linear Algebra Lecture 1 36 / 36

Parametric Solution set

Given the RREF of the augmented matrix of the system, write the solution
set explicitly (using free variables, if any).
" #
1 0 0 1 0
1
0 1 1 3 1
SS = {( x4 , 1 + x3 3x4 , x3 , x4 ) | x3 , x4 R}
" #
1 0 0 0 4
2 SS = {( , , x3 , x4 ) | y , z R}
0 1 0 0 11

Math 114Introduction to Linear Algebra Lecture 1 36 / 36

Parametric Solution set

Given the RREF of the augmented matrix of the system, write the solution
set explicitly (using free variables, if any).
" #
1 0 0 1 0
1
0 1 1 3 1
SS = {( x4 , 1 + x3 3x4 , x3 , x4 ) | x3 , x4 R}
" #
1 0 0 0 4
2 SS = {( 4, , x3 , x4 ) | y , z R}
0 1 0 0 11

Math 114Introduction to Linear Algebra Lecture 1 36 / 36

Parametric Solution set

Given the RREF of the augmented matrix of the system, write the solution
set explicitly (using free variables, if any).
" #
1 0 0 1 0
1
0 1 1 3 1
SS = {( x4 , 1 + x3 3x4 , x3 , x4 ) | x3 , x4 R}
" #
1 0 0 0 4
2 SS = {( 4, 11, x3 , x4 ) | y , z R}
0 1 0 0 11

Math 114Introduction to Linear Algebra Lecture 1 36 / 36