Basis for a Subspace

Definition:

Question: How can we use the RREF to find a basis for the column space of a matrix?

Example: Find a basis for the column space of

1 1 1 3

2 1 3 2

1 2 4 1

3 0 2 5

Example: Find bases for the null space and column space of

1 1 1 3

2 1 3 2

1 2 4 1

3 0 2 5

Example: Show that 111

,121

,212

is a basis for

. (Hint: Recall the special situation. n vectors give a basis for if and only if they are linearly independent.)

Example: Find a basis for the null space of the matrix

1 3 0 1 1

1 2 1 2 3

Also give a basis for the column space of the matrix.

Example: Find a basis for the subspace spanned by the vectors

1123

,

2314

,

0132

,

1477

,

3769

Section 2.9

Dimension of a Subspace

Theorem: (The Basis Theorem) Any two basis for a subspace H of have the same number of elements. Furthermore, if a basis for H has p elements, then any linearly independent subset of H with p elements is a basis for H, and any subset of H with p entries that spans H is a basis for H.

Corollary: A subset of is a basis for if and only if it consists of linearly independent vectors.

Terminology: Standard basis for .

Example: Show that

2 32

2 2| , , ∈

is a subspace of . Give a basis for , and the dimension of .

Finally, determine whether 111

∈ .

Nullity and Rank

Theorem: (rank theorem) If is an matrix, then

rank A + nullity A = n

Rank and the Invertible Matrix Theorem

Theorem: (The Invertible Matrix Theorem) Suppose A is an matrix. Then the following statements are equivalent to

A being an invertible matrix.

m. The columns of form a basis for . n. Col o. dimCol p. rank q. Nul 0 r. dimNul 0

Example: A basis for the null space of a 2 by 3 matrix A is given by

1

1

2

Give the rank of A.

Example: The RREF of the matrix A is

1 1 0

0 0 1

0 0 0

Give the rank of A, the nullity of A, and a basis for the null space of A.

B-Coordinates of a Vector x in a Subspace H with Respect to a Basis B for H

Definition:

Example: Show that 21, 11

is a basis for . Then

give 11

.

Exam

mple: Prooblem 30

0 in the ttextbookk.