Fundamental Subspaces Associated with a Matrix

Introduced in Math 115

Note

Let $A\in M_{mxn}(\mathbb{R})$. The nullspace of A (sometimes called the kernel of A) is the subset of ${\mathbb{R}}^n$ defined by

Null(A) = $\{\vec{x}\in {\mathbb{R}}^n\mid A\vec{x}=\vec{0}\}$

[!Column Space]

Let $A=[{\vec{a}}_1\dots {\vec{a}}_n]\in M_{mxn}(\mathbb{R})$. The column space of A is the subset of ${\mathbb{R}}^m$ defined by

Col(A) = $\{A\vec{x}\mid \vec{x}\in {\mathbb{R}}^n\}$ = Span$\{{\vec{a}}_1,\dots ,{\vec{a}}_n\}$

[!Theorem 4.6.3]

Let $A\in M_{mxn}(\mathbb{R})$. Then Null(A) is a subspace of ${\mathbb{R}}^n$ and Col(A) is a subspace of ${\mathbb{R}}^m$

Note

Let $A\in M_{mxn}(\mathbb{R})$ The nullity of A, denoted by nullity(A) is defined by

nullity(A) = n - rank(A)

[!Theorem 4.6.8]

Let $A\in M_{mxn}(\mathbb{R})$ Then

• dim(Null(A)) = nullity(A)
• dim(Col(A)) = rank(A)

[!Rank-Nullity Theorem]

For any $A\in M_{mxn}(\mathbb{R})$

rank(A) + nullity(A) = n