Fundamental Spaces of a Matrix: Row Space, Column Space, and Null Space - Prof. Donald O. , Study notes of Mathematics

Download Fundamental Spaces of a Matrix: Row Space, Column Space, and Null Space - Prof. Donald O. and more Study notes Mathematics in PDF only on Docsity!

Dimension and Structure

7.3 The Fundamental Spaces of a Matrix

October 30, 2009

Three spaces associated with an m × n matrix A, are

1. the row space of A, denoted by row(A). It is the subspace of Rn that is spanned by the row vectors of A.
2. the column space of A, denoted by col(A). It is the subspace of Rm^ that is spanned by the column vectors of A.
3. the null space of A, denoted by null(A). It is the solution space of Ax = 0. (Recall it is a subspace of Rn).

Definition 7.3.

The dimension of the row space of a matrix A is called ther rank of A and is denoted by rank(A); and the dimension of the null space of A is called the nullity of A and is denoted by nullity(A).

Example

Describe the row spaces for

(a) A =

 (^) A basis for row(A) is

B = {(1, 0 , 1 , 0)T^ , (0, 1 , 23 , 13 )T^ }.

(b) B =

[

]

A basis for row(B) is B = {(1, 2 , 0)T^ , (0, 0 , 1)T^ }.

Defintion: Orthogonal Complement

Let U be a subspace of Rn. The orthogonal complement of U, denoted by U⊥, is the set of all vectors v in Rn^ such that v ⊥ u for every vector u in U.

Theorem

Let U be a subspace of Rn. Then

1. U⊥^ is a subspace of Rn.
2. The intersection of the subspaces U and U⊥^ is the zero subspace; that is, U⊥^ ∩ U = { 0 }.
3. (U⊥)⊥^ = U.

Theorem 7.3.

If A is an m × n matrix, then the row space of A and the null space of A are orthogonal complements.

Theorem 7.3.

If A is an m × n matrix, then the column space of A and the null space of AT^ are orthogonal complements.

Theorem 7.3.

If A and B are matrices with the same number of columns, then the following statements are equivalent.

(a) A and B have the same row space.

(b) A and B have the same null space.

(c) The row vectors of A are linear combinations of the row vectors of B, and conversely.