Math 107 (Spring 2022)

Week 1: Lecture 1

Emre Mengi

February 11, 2022

What is Linear Algebra and Why?

1
 What is Linear Algebra and Why?

Linear algebra studies linear objects

1
 What is Linear Algebra and Why?

Linear algebra studies linear objects such as

• linear equations,

1
 What is Linear Algebra and Why?

Linear algebra studies linear objects such as

• linear equations,

• linear functions,

1
 What is Linear Algebra and Why?

Linear algebra studies linear objects such as

• linear equations,

• linear functions,

• linear sets.

1
 What is Linear Algebra and Why?

Linear algebra studies linear objects such as

• linear equations,

• linear functions,

• linear sets.

Why linear objects?

1
 What is Linear Algebra and Why?

Linear algebra studies linear objects such as

• linear equations,

• linear functions,

• linear sets.

Why linear objects?

• We can say a lot about them; there are common tools (a general theory)
that apply to all linear objects.

1
 What is Linear Algebra and Why?

Linear algebra studies linear objects such as

• linear equations,

• linear functions,

• linear sets.

Why linear objects?

• We can say a lot about them; there are common tools (a general theory)
that apply to all linear objects.

• Applications give rise to linear objects.

1
 What is Linear Algebra and Why?

Linear algebra studies linear objects such as

• linear equations,

• linear functions,

• linear sets.

Why linear objects?

• We can say a lot about them; there are common tools (a general theory)
that apply to all linear objects.

• Applications give rise to linear objects.

• Nonlinear objects can many times be approximated by linear objects well.

1
 What is Linear Algebra and Why?

Linear algebra studies linear objects such as

• linear equations,

• linear functions,

• linear sets.

Why linear objects?

• We can say a lot about them; there are common tools (a general theory)
that apply to all linear objects.

• Applications give rise to linear objects.

• Nonlinear objects can many times be approximated by linear objects well.

• At other times, linear objects may give insight about nonlinear objects.

1
 Systems of Linear Equations

A system of 2 linear equations in 2 variables

`1 : a11 x1 + a12 x2 = b1
`2 : a21 x1 + a22 x2 = b2

2
 Systems of Linear Equations

A system of 2 linear equations in 2 variables

`1 : a11 x1 + a12 x2 = b1
`2 : a21 x1 + a22 x2 = b2

given — a11 , a12 , a21 , a22 , b1 , b2 , variables — x1 , x2

2
 Systems of Linear Equations

A system of 2 linear equations in 2 variables

`1 : a11 x1 + a12 x2 = b1
`2 : a21 x1 + a22 x2 = b2

given — a11 , a12 , a21 , a22 , b1 , b2 , variables — x1 , x2

-2

-4

-6
-6 -4 -2 0 2 4 6

2
 Systems of Linear Equations

A system of 2 linear equations in 2 variables

`1 : a11 x1 + a12 x2 = b1
`2 : a21 x1 + a22 x2 = b2

given — a11 , a12 , a21 , a22 , b1 , b2 , variables — x1 , x2

-2

-4

-6
-6 -4 -2 0 2 4 6

• A solution (s1 , s2 ) when substituted for (x1 , x2 ) satisfies both linear equations.

2
 Systems of Linear Equations

A system of 2 linear equations in 2 variables

`1 : a11 x1 + a12 x2 = b1
`2 : a21 x1 + a22 x2 = b2

given — a11 , a12 , a21 , a22 , b1 , b2 , variables — x1 , x2

-2

-4

-6
-6 -4 -2 0 2 4 6

• A solution (s1 , s2 ) when substituted for (x1 , x2 ) satisfies both linear equations.
(for example, intersection point of `1 and `2 in the plot above is the solution)

2
 Systems of Linear Equations

Examples

(1)
`1 : 2x1 + x2 = 1
`2 : x1 + 2x2 = 1

has a unique solution (1/3, 1/3) satisfying both `1 and `2 .

3
 Systems of Linear Equations

Examples

(1)
`1 : 2x1 + x2 = 1
`2 : x1 + 2x2 = 1

has a unique solution (1/3, 1/3) satisfying both `1 and `2 .

(2)
`1 : 2x1 + x2 = 1
`2 : 4x1 + 2x2 = 4

has no solution.

3
 Systems of Linear Equations

Examples

(1)
`1 : 2x1 + x2 = 1
`2 : x1 + 2x2 = 1

has a unique solution (1/3, 1/3) satisfying both `1 and `2 .

(2)
`1 : 2x1 + x2 = 1
`2 : 4x1 + 2x2 = 4

has no solution.

(3)
`1 : 2x1 + x2 = 1
`2 : 4x1 + 2x2 = 2

has infinitely many solutions; any (s1 , s2 ) satisfying `1 is a solution.

3
 Systems of Linear Equations

`1 : a11 x1 + a12 x2 = b1
`2 : a21 x1 + a22 x2 = b2

4
 Systems of Linear Equations

`1 : a11 x1 + a12 x2 = b1
`2 : a21 x1 + a22 x2 = b2

Possibilities Regarding the Number of Solutions

(1) Unique solution (`1 , `2 have different slopes)

(2) No solution (`1 , `2 are parallel)

(3) Infinitely many solutions (`1 , `2 are identical)

4
 Systems of Linear Equations

A linear equation in n variables

a1 x 1 + a2 x 2 + · · · + an x n = b

variables — x1 , x2 , . . . , xn .

5
 Systems of Linear Equations

A linear equation in n variables

a1 x 1 + a2 x 2 + · · · + an x n = b

variables — x1 , x2 , . . . , xn .

Geometric View

5
 Systems of Linear Equations

A linear equation in n variables

a1 x 1 + a2 x 2 + · · · + an x n = b

variables — x1 , x2 , . . . , xn .

Geometric View

All such (x1 , . . . , xn ) satisfying the linear equation form

• a line when n = 2,

5
 Systems of Linear Equations

A linear equation in n variables

a1 x 1 + a2 x 2 + · · · + an x n = b

variables — x1 , x2 , . . . , xn .

Geometric View

All such (x1 , . . . , xn ) satisfying the linear equation form

• a line when n = 2,

• a plane when n = 3,

5
 Systems of Linear Equations

A linear equation in n variables

a1 x 1 + a2 x 2 + · · · + an x n = b

variables — x1 , x2 , . . . , xn .

Geometric View

All such (x1 , . . . , xn ) satisfying the linear equation form

• a line when n = 2,

• a plane when n = 3,

• a hyperplane when n 4.

5
 Systems of Linear Equations

A linear equation in n variables

a1 x 1 + a2 x 2 + · · · + an x n = b

variables — x1 , x2 , . . . , xn .

Geometric View

All such (x1 , . . . , xn ) satisfying the linear equation form

• a line when n = 2,

• a plane when n = 3,

• a hyperplane when n 4.

Examples

2x1 x2 = 3
x1 + 5x2 + x3 = 2

5
 Systems of Linear Equations

A system of m linear equations in n variables

a11 x1 + a12 x2 + · · · + a1n xn = b1

a21 x1 + a22 x2 + · · · + a2n xn = b2

..
.

am1 x1 + am2 x2 + · · · + amn xn = bm

6
 Systems of Linear Equations

A system of m linear equations in n variables

a11 x1 + a12 x2 + · · · + a1n xn = b1

a21 x1 + a22 x2 + · · · + a2n xn = b2

..
.

am1 x1 + am2 x2 + · · · + amn xn = bm

given — aij , bi ( for all i 2 {1, . . . , m} and j 2 {1, . . . , n} )

variables — x1 , . . . , xn

6
 Systems of Linear Equations

A system of m linear equations in n variables

a11 x1 + a12 x2 + · · · + a1n xn = b1

a21 x1 + a22 x2 + · · · + a2n xn = b2

..
.

am1 x1 + am2 x2 + · · · + amn xn = bm

given — aij , bi ( for all i 2 {1, . . . , m} and j 2 {1, . . . , n} )

variables — x1 , . . . , xn

A solution (s1 , . . . , sn ) when substituted for (x1 , . . . , xn ) satisfies

all m linear equations.

6
 Systems of Linear Equations

A system of m linear equations in n variables

a11 x1 + a12 x2 + · · · + a1n xn = b1

a21 x1 + a22 x2 + · · · + a2n xn = b2

..
.

am1 x1 + am2 x2 + · · · + amn xn = bm

given — aij , bi ( for all i 2 {1, . . . , m} and j 2 {1, . . . , n} )

variables — x1 , . . . , xn

A solution (s1 , . . . , sn ) when substituted for (x1 , . . . , xn ) satisfies

all m linear equations.

Geometric View
A solution of the system above is an intersection point of m hyperplanes.

6
 Matrix Representations

a11 x1 + a12 x2 + · · · + a1n xn = b1

a21 x1 + a22 x2 + · · · + a2n xn = b2

..
.

am1 x1 + am2 x2 + · · · + amn xn = bm

7
 Matrix Representations

a11 x1 + a12 x2 + · · · + a1n xn = b1

a21 x1 + a22 x2 + · · · + a2n xn = b2

..
.

am1 x1 + am2 x2 + · · · + amn xn = bm

Coefficient Matrix (associated with the system above)

2 3
a11 a12 . . . a1n
6 a 7
6 21 a22 . . . a2n 7
6 .. . 7
4 . .. 5
am1 am2 . . . amn

7
 Matrix Representations

a11 x1 + a12 x2 + · · · + a1n xn = b1

a21 x1 + a22 x2 + · · · + a2n xn = b2

..
.

am1 x1 + am2 x2 + · · · + amn xn = bm

Coefficient Matrix (associated with the system above)

2 3
a11 a12 . . . a1n
6 a 7
6 21 a22 . . . a2n 7
6 .. . 7
4 . .. 5
am1 am2 . . . amn

(an m ⇥ n matrix; has m rows, n columns)

7
 Matrix Representations

a11 x1 + a12 x2 + · · · + a1n xn = b1

a21 x1 + a22 x2 + · · · + a2n xn = b2

..
.

am1 x1 + am2 x2 + · · · + amn xn = bm

Coefficient Matrix (associated with the system above)

2 3
a11 a12 . . . a1n
6 a 7
6 21 a22 . . . a2n 7
6 .. . 7
4 . .. 5
am1 am2 . . . amn

(an m ⇥ n matrix; has m rows, n columns)

Augmented Matrix (associated with the system above)

2 3
a11 a12 . . . a1n b1
6 a 7
6 21 a22 . . . a2n b2 7
6 .. . 7
4 . .. 5
am1 am2 . . . amn bm

7
 Matrix Representations

a11 x1 + a12 x2 + · · · + a1n xn = b1

a21 x1 + a22 x2 + · · · + a2n xn = b2

..
.

am1 x1 + am2 x2 + · · · + amn xn = bm

Coefficient Matrix (associated with the system above)

2 3
a11 a12 . . . a1n
6 a 7
6 21 a22 . . . a2n 7
6 .. . 7
4 . .. 5
am1 am2 . . . amn

(an m ⇥ n matrix; has m rows, n columns)

Augmented Matrix (associated with the system above)

2 3
a11 a12 . . . a1n b1
6 a 7
6 21 a22 . . . a2n b2 7
6 .. . 7
4 . .. 5
am1 am2 . . . amn bm

(an m ⇥ (n + 1) matrix; has m rows, n + 1 columns)

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 Matrix Representations

Coefficient Matrix
2 3
a11 a12 . . . a1n
6 a 7
6 21 a22 . . . a2n 7
6 .. .. 7
4 . . 5
am1 am2 . . . amn

Augmented Matrix
2 3
a11 a12 . . . a1n b1
6 a 7
6 21 a22 . . . a2n b2 7
6 .. .. 7
.
4 . 5
am1 am2 . . . amn bm

Example

2x1 + 5x2 = 3
3x1 x2 = 4

8
 Matrix Representations

Coefficient Matrix
2 3
a11 a12 . . . a1n
6 a 7
6 21 a22 . . . a2n 7
6 .. .. 7
4 . . 5
am1 am2 . . . amn

Augmented Matrix
2 3
a11 a12 . . . a1n b1
6 a 7
6 21 a22 . . . a2n b2 7
6 .. .. 7
.
4 . 5
am1 am2 . . . amn bm

Example

2x1 + 5x2 = 3
3x1 x2 = 4

Corresponding coefficient matrix

" #
2 5
3 1

8
 Matrix Representations

Coefficient Matrix
2 3
a11 a12 . . . a1n
6 a 7
6 21 a22 . . . a2n 7
6 .. .. 7
4 . . 5
am1 am2 . . . amn

Augmented Matrix
2 3
a11 a12 . . . a1n b1
6 a 7
6 21 a22 . . . a2n b2 7
6 .. .. 7
.
4 . 5
am1 am2 . . . amn bm

Example

2x1 + 5x2 = 3
3x1 x2 = 4

Corresponding coefficient matrix and augmented matrix

" # " #
2 5 2 5 3
and
3 1 3 1 4

8
 Temperature Example

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

9
 Temperature Example

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2
Temperatures T1 , T2 , T3 in mesh above are given by the average of the adjacent nodes.
0.1

0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

9
 Temperature Example

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2
Temperatures T1 , T2 , T3 in mesh above are given by the average of the adjacent nodes.
0.1

0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

• Set up a system of linear equations whose solution yields T1 , T2 , T3 .

9
 Temperature Example

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2
Temperatures T1 , T2 , T3 in mesh above are given by the average of the adjacent nodes.
0.1

0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

• Set up a system of linear equations whose solution yields T1 , T2 , T3 .

• Write down the augmented matrix corresponding to the linear system.

9
 1 Temperature Example
0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2
20 + 50 + T2 + T3
T1 =
0.1 4

0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

10
 1 Temperature Example
0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2
20 + 50 + T2 + T3
T1 =
0.1 4
20 + 30 + T1 + T3
0 T2 =
0 0.1 0.2 0.3 4
0.4 0.5 0.6 0.7 0.8 0.9 1

10
 1 Temperature Example
0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2
20 + 50 + T2 + T3
T1 =
0.1 4
20 + 30 + T1 + T3
0 T2 =
0 0.1 0.2 0.3 4
0.4 0.5 0.6 0.7 0.8 0.9 1
30 + 50 + T1 + T2
T3 =
4

10
 1 Temperature Example
0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2
20 + 50 + T2 + T3
T1 =
0.1 4
20 + 30 + T1 + T3
0 T2 =
0 0.1 0.2 0.3 4
0.4 0.5 0.6 0.7 0.8 0.9 1
30 + 50 + T1 + T2
T3 =
4

System of Linear Equations

4T1 T2 T3 = 70

T1 + 4T2 T3 = 50 ,

T1 T2 + 4T3 = 80

10
 1 Temperature Example
0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2
20 + 50 + T2 + T3
T1 =
0.1 4
20 + 30 + T1 + T3
0 T2 =
0 0.1 0.2 0.3 4
0.4 0.5 0.6 0.7 0.8 0.9 1
30 + 50 + T1 + T2
T3 =
4

System of Linear Equations & Corresponding Augmented Matrix

4T1 T2 T3 = 70 2 3
4 1 1 70
6 7
T1 + 4T2 T3 = 50 , 4 1 4 1 50 5
1 1 4 80
T1 T2 + 4T3 = 80

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 Summary

• A Linear Equation

• A System of m Linear Equations in n Variables

• Possibilities Regarding the Number of Solutions of a Linear System of Equations

• Coefficient and Augmented Matrices

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