TITLE: How is the column space of a matrix A orthogonal to its nullspace?
QUESTION [18 upvotes]: How do you show that the column space of a matrix A is orthogonal to its nullspace?

REPLY [6 votes]: The OP's question would have been well-phrased had he specified that the matrix $A$ is symmetric i.e. $A=A^\top$, in which case ${\rm colspan}(A)={\rm rowspan}(A^\top)={\rm rowspan}(A)$.  Now, consider the definition of ${\rm null}(A)$ as the space of all vectors $\mathbf{v}$ such that $A\mathbf{v}=\mathbf{0}$.  Letting $\mathbf{a}_1,\ldots,\mathbf{a}_n$ be the rows (and columns) of $A$, matrix multiplication tells us that $\mathbf{a}_i\cdot\mathbf{v}=0$ for each $i=1,\ldots,\dim(A)$.  Thus any vector $\mathbf{v}\in{\rm null}(A)$ is orthogonal to ${\rm colspan}(A)$.  It follows that ${\rm null}(A)\perp{\rm colspan}(A)$.$\square$