Linear Algebra: Subspaces, Basis, Dimension, and Rank, Study notes of Linear Algebra

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Math 215: Introduction to Linear Algebra

Section 3.5: Subspaces, Basis, Dimension, and

Rank

Paul Dostert

September 2, 2008

Subspaces

2 / 12

The motivation for subspaces comes from the question: What does thespace spanned by

[

0]

and

[

0]

look like? Is this in

R

2

or

R

3

A

subspace

of

R

n

is any collection

S

of vectors in

R

n

st

The zero vector

is in

S

If

u

v

S

, then

u

v

S

S

is closed under addition)

If

u

S

and

c

is a scalar, then

c

u

S

S

is closed under scalar

multiplication)

Ex

: Show that the space spanned by

[

0]

and

[

0]

is a subspace of

R

3

Thm

: Let

v

1

v

k

be vectors in

R

n

. Then

span

v

1

v

k

is a subspace

of

R

n

. We say

span

v

1

v

k

is

the subspace spanned by

v

1

v

k

Ex

: Do vectors of the form

[

x^1

]

form a subspace of

R

2

? Do vectors of the

form

x y

x

y

form a subspace of

R

3

Matrix subspaces & null space

Thm

: Let

B

be any matrix that is row equivalent to a matrix

A

. Then

row

B

row

A

Thm

: Let

A

be an

m

×

n

matrix and let

N

be the set of solutions of the

homogeneous linear system

A

x

. Then

N

is a subspace of

R

n

Ex

: Prove this theorem.

Let

A

be an

m

×

n

matrix. The

null space

of

A

is the subspace of

R

n

consisting of solutions of the homogeneous linear system

A

x

. It is

denoted by

null

A

The idea of null space comes into the proof of the following VERY importanttheorem. Thm

: Let

A

be a matrix whose entries are real numbers. For any system of

linear equations

A

x

b

, exactly one of the following is true:

(a)

There is no solution.

(b)

There is a unique solution.

(c)

There are infinitely many solutions.

Pf

: The idea is that we show if we cannot have (a) or (b) then (c) must be

true. To show (c), we show the null space has infinitely many vectors.

Basis

5 / 12

A

basis

for a subspace

S

of

R

n

is a set of vectors in

S

that are linearly

independent and span

S

We call the standard unit vectors

e

1

e

n

R

n

the

standard basis

for

R

n

Ex

: Find a basis for

span

Ex

: Find a basis for

span

Ex

: Find a basis for the row space of

A

Dimension

Thm

(The Basis Thm): Let

S

be a subspace of

R

n

. Then any two bases for

S

have the same number of vectors.

Ex

: Describe how you would attempt to prove this.

If

S

is a subspace of

R

n

, then the number of vectors in a basis for

S

is called

the

dimension

of

S

, denoted

dim

S

Note: We define

dim

Ex

: For the matrix on the previous slide,

A

, what are

the dimensions of the row, column and null spaces? How are these related? Thm

: The row and column spaces of a matrix

A

have the same dimension.

Ex

: Look at the proof. The idea is

dim

row

A

dim

row

R

nonzero rows of

R

leading 1s of

R

dim

col

R

dim

col

A

Rank

The

rank

of a matrix

A

is the dimension of its row and column spaces and

is denoted by

rank

A

The

nullity

of a matrix

A

is the dimension of its null space and is denoted

by

nullity

A

Cor

: For any matrix

A

rank

A

T

rank

A

Ex

: Prove this.

Thm

(The Rank Thm): If

A

is an

m

×

n

matrix, then

rank

A

nullity

A

n.

Ex

: Find the nullity of

A

and

B

by using

the rank theorem (and NOT by solving

A

x

Using the Fundamental Thm

10 / 12

Thm

: Let

A

be an

m

×

n

matrix. Then

rank

A

T

A

rank

A

and

A

T

A

is

invertible iff

rank

A

n

Ex

: Prove using the rank theorem and the fund. thm.

Ex

: Show that the vectors

, and

form a basis for

R

3

Recall that is

B

is a basis for a subspace

S

of

R

n

then we can write any

vector in

S

as a linear combination of basis vectors. The following takes this

idea one step further. Thm

: Let

S

be a subspace of

R

n

and

B

v

1

v

k

a basis for

S

. Every

v

S

can be written as a unique linear combination of basis vectors in

B

v

c

1

v

1

c

k

v

k

with unique

c

i

Ex

: Prove this.

Coordinate Representation

Let

S

be a subspace of

R

n

and let

B

v

1

v

k

be a basis for

S

. Let

v

c

1

v

1

c

k

v

k

be a vector in

S

. Then the

c

i

are called the

coordinates of

v

with respect to

B

, and the column vector

[

v

]

B

c

c

k

is called the

coordinate vector of

v

with respect to

B

Ex

: Show that

w

is in

span

B

and find the coordinate vector

[

w

]

B

for

B

w