Question

Linear Algebra

Explain why the nullspace of a matrix A is always nonempty.

What is the definition of the column space of a matrix A? Briefly explain why this is different from the nullspace.

Answer 1

The nullspace of $m\times n$ matrix is the space $\mbox{null}(A)=\{x\in\mathbb R^n:Ax=0\}$ . Since $A0=0$ , we have  $0\in\mbox{null}(A)$. Thus, $\mbox{null}(A)$ is always non-empty.

Let $A_1,\cdots,A_n$ be the columns of the matrix . Then its column space is

$\mbox{column}(A)=\{x_1A_1+\cdots+x_nA_n:x_1,\cdots,x_n\in\mathbb R\}$

For $m\times n$ matrix, nullspace is a subspace of $\mathbb R^n$ but columnspace is subspace of $\mathbb R^m$ . Thus, they are different in general