Lecture 9 recap
1) Two properties that a linear span must have.
2) Linear span inside linear span theorem.
3) 'Useless' vector
4) Definition of subspace.
5) How to show a subset is not a subspace using 1).
6) How to show a subset is a subspace by writing as
linear span.
Lecture 10
Subspaces (cont’d)
Linear independence
Geometrical examples
2 3
Let u be a nonzero vector in or .
span{u} is the set of all linear combinations (or scalar
multiples) of u.
Geometrically, span{u} is a straight line passing through
the origin.
u
1.5u
origin
2u
Geometrical examples
2 3
Let u be a nonzero vector in or .
(In 2
) u (u1 ,u2 ), span{u} {(cu1 ,cu2 )|c }
(explicit representation)
(implicit representation i.e. equation of line?)
u
(u1 ,u2 )
origin
Geometrical examples
2 3
Let u be a nonzero vector in or .
(In 2
) u (u1 ,u2 ), span{u} {(cu1 ,cu2 )|c }
(explicit representation)
(implicit representation i.e. equation of line?)
Remember that a line in 3
u
cannot be represented by a (u1 ,u2 ,u3 )
single linear equation. origin
Geometrical examples
2 3
Let u, v be two nonzero vectors in or .
span{u, v} is the set of all linear combinations of u and v.
{ su t v | s , t }
If u and v are not parallel,
su t v , s , t 0
su t v , s 0 , t 0 v
u
origin
su t v , s , t 0 su t v , s 0 , t 0
Geometrical examples
2 3
Let u, v be two nonzero vectors in or .
span{u, v} is the set of all linear combinations of u and v.
{ su t v | s , t }
If u and v are not parallel, span{u, v} is a plane containing
v the origin.
u
origin
Geometrical examples
2 3
Let u, v be two nonzero vectors in or .
span{u, v} is the set of all linear combinations of u and v.
{ su t v | s , t }
What if u and v are parallel?
span{ u, v} span{u}
straight line passing
through the origin.
Geometrical examples
If u and v are not parallel,
(In 2
) span{u, v} 2
.
(In 3
) span{u, v} { su tv | s , t } (explicit representation)
(implicit representation, i.e. equation of the plane?)
u (u1 , u2 , u3 ), v (v1 , v2 , v3 )
v
u
origin
Remark (All subspaces of R2)
2
The following are all the subspaces of :
Remark (All subspaces of R3)
3
The following are all the subspaces of :
Theorem (Solution set of homogeneous
systems)
The solution set of a homogeneous system of linear
n
equations in n variables is a subspace of .
a11 x1 a12 x2 ... a1n xn 0
a x a x ... a x 0
21 1 22 2 2n n
: : : :
am1 x1 am 2 x2 ... amn xn 0
Example
Investigate the solution set of the following homogeneous
linear system:
x 2 y 3z 0
2 x 4 y 6 z 0
3x 6 y 9 z 0
1 2 3 0 1 2 3 0
2 4 6 0 Gaussian 0 0 0 0
Elimination
3 6 9 0 0 0 0 0
Example
1 2 3 0
0 0 0 0
0 0 0 0
3
Geometrically, the solution set is a plane in containing
the origin.
Example
Investigate the solution set of the following homogeneous
linear system:
x 2 y 3z 0
2 x 4 y 6 z 0
3 x 7 y 8 z 0
1 2 3 0 1 0 5 0
2 4 6 0 Gaussian 0 1 1 0
3 7 8 0 Elimination 0 0 0 0
Example
1 0 5 0
0 1 1 0
0 0 0 0
3
Geometrically, the solution set is a line in passing
through the origin.
Example
Investigate the solution set of the following homogeneous
linear system:
x 2 y 3z 0
4x y 2z 0
3 x 7 y 8 z 0
1 2 3 0 1 0 0 0
4 1 2 0 Gaussian 0 1 0 0
3 7 8 0 Elimination 0 0 1 0
The solution set is
the zero space {0}.
Abstract definition of subspace
n
Let V be a non-empty subset of .
n
Then V is a subspace of if and only if
for all u, v V and c , d , cu dv V .
n
V
Discussion on redundancy
If uk is a linear combination of u1 ,u2 ,...,uk-1 , then
span{u1 ,u2 ,...,uk-1} = span{u1 ,u2 ,...,uk-1 ,uk }
We say that uk is redundant in the span of {u1 , u2 ,..., uk 1 ,uk }.
I am redundant
Having me around does uk
not 'add value'
Definition (linear independence)
Let S { u1 , u2 ,..., uk } n . Consider the solutions to the
following equation (values of c1 , c2 ,..., ck )
c1u1 c2 u2 ... ck uk 0 (*)
1) Clearly, c1 0, c2 0,..., ck 0 is a solution. This is called
the trivial solution to (*).
2) S is called a linearly independent set if (*) has only
the trivial solution. In this case, we say that u1 , u2 ,..., uk
are linearly independent vectors.
Definition (linear independence)
Let S { u1 , u2 ,..., uk } n . Consider the solutions to the
following equation (values of c1 , c2 ,..., ck )
c1u1 c2 u2 ... ck uk 0 (*)
2) S is called a linearly independent set if (*) has only
the trivial solution. In this case, we say that u1 , u2 ,..., uk
are linearly independent vectors.
3) S is called a linearly dependent set if (*) has
non-trivial solutions. In this case, we say that u1 , u2 ,..., uk
are linearly dependent vectors.
Definition (linear independence)
Let S { u1 , u2 ,..., uk } n . Consider the solutions to the
following equation (values of c1 , c2 ,..., ck )
c1u1 c2 u2 ... ck uk 0 (*) It's all
about linear
(Only) trivial
combinations...
solution to (*)??
What does it mean?
Example (linear independence)
Determine whether (1, 2, 3),(5, 6, 1),(3, 2,1) are linearly
3
independent vectors in .
Vector equation:
Linear system:
Example (linear independence)
Determine whether (1, 2, 3),(5, 6, 1),(3, 2,1) are linearly
3
independent vectors in .
Solving linear system:
1 5 3 0 1 5 3 0
2 6 2 0 Gaussian 0 16 8 0
Elimination
3 1 1 0 0 0 0 0
How many solutions a 5b 3c 0
does the linear 2a 6b 2c 0
3a b c 0
system have?
Example (linear independence)
Determine whether (1, 2, 3),(5, 6, 1),(3, 2,1) are linearly
3
independent vectors in . The vectors are
Solving linear system: linearly dependent.
1 5 3 0 1 5 3 0
2 6 2 0 Gaussian 0 16 8 0
Elimination
3 1 1 0 0 0 0 0
How many solutions
does the equation a(1, 2, 3) b(5, 6, 1) c(3, 2,1) (0, 0, 0)
have?
Example (linear independence)
Determine whether (1, 0, 0,1),(0, 2,1, 0),(1, 1,1,1) are linearly
4
independent vectors in .
Vector equation:
Linear system:
Linear independence: 1 or 2 vectors
S {u}. When is S a linearly independent set?
When does the equation cu 0 have only the
trivial solution c 0?
S {u} is a linearly independent set if and only if
u 0.
Linear independence: 1 or 2 vectors
S {u, v}. When is S a linearly independent set?
When does the equation c1u c2v 0 have non trivial
solutions for c1 and c2 ?
S {u, v} is a linearly dependent set if and only if
u and v are scalar multiples of each other.
What if a set contains the zero vector?
Let S be a finite set of vectors from n . Prove that if
0 S , then S is a linearly dependent set.
Proof:
Theorem (another way to look at
linear independence)
Recall the discussion on redundancy.
Let S {u1 , u2 ,..., uk } be a set of vectors in n
, where k 2.
1) S is linearly dependent if and only if at least one ui S
can be written as a linear combination of the other
vectors in S , that is,
ui a1u1 a2 u2 ... ai1ui1 ai1ui1 ... ak uk
for some a1 ,..., ai1 , ai1 ,..., ak .
Theorem (another way to look at
linear independence)
Recall the discussion on redundancy.
Let S {u1 , u2 ,..., uk } be a set of vectors in n
, where k 2.
2) S is linearly independent if and only if no vector in S
can be written as a linear combination of the other
vectors in S .
Remark
So a set a vectors is linearly dependent implies that
there exists at least one 'redundant' vector in the set.
A set a vectors is linearly independent implies that
there is no 'redundant' vector in the set.
Example
S {(2, 4),(1, 0),(0, 3)}. Is S a linearly independent set?
S {(1, 0, 0),(0, 2, 0),(0, 0, 5)}. Is S a linearly independent set?
Theorem (guaranteed dependence)
Let S {u1 , u2 ,..., uk } be a set of vectors in n
.
If k n, then S is linearly dependent.
Example (guaranteed dependence)
2
1) A set of three or more vectors in is always linearly
dependent.
3
2) A set of four or more vectors in is always linearly
dependent.
Linear independence (geometrical)
2 3
For two vectors in or , recall the following:
S {u, v} is a linearly dependent set if and only if
u and v are scalar multiples of each other (they lie on
the same line).
v
v
u u u
v
Linearly Linearly Linearly
dependent dependent independent
Linear independence (geometrical)
3
For three vectors in :
S {u, v , w} is a linearly dependent set if and only if
they lie on the same line or the same plane (when their
initial points are placed at the origin).
v
u u u, v , w lie on the
same line
origin w origin
{u} is a linearly {u, v} is a linearly {u, v , w} is a linearly
independent set dependent set dependent set
Linear independence (geometrical)
3
For three vectors in :
S {u, v , w} is a linearly dependent set if and only if
they lie on the same line or the same plane (when their
initial points are placed at the origin).
v
u, v , w lie on the
u u
same plane
origin origin w
{u} is a linearly {u, v} is a linearly {u, v , w} is a linearly
independent set dependent set dependent set
Linear independence (geometrical)
3
For three vectors in :
S {u, v , w} is a linearly dependent set if and only if
they lie on the same line or the same plane (when their
initial points are placed at the origin).
span{ u, v} u span{u, v}u
w
u w v v
origin origin origin
{u} is a linearly {u, v} is a linearly {u, v , w} is a linearly
independent set independent set dependent set
Linear independence (geometrical)
3
For three vectors in :
S {u, v , w} is a linearly dependent set if and only if
they lie on the same line or the same plane (when their
initial points are placed at the origin).
w span{ u, v}
span{u, v} u span{u, v}u
u
v v
origin origin origin
{u} is a linearly {u, v} is a linearly {u, v , w} is a linearly
independent set independent set independent set
End of Lecture 10
Lecture 11:
Linear independence (cont’d)
Bases
Dimensions (till Example 3.6.6)
