## Franklin's Notes

### Matrix-vector multiply

A basic matrix-vector multiply takes the form where $A\in\mathbb C^{m\times n}$, $x\in\mathbb C^n$, and $b\in\mathbb C^m$. The entries of $b$ are given by for $i\in {1,...,m}$. Another interpretation of the matrix-vector multiply is that it expresses $b$ as a linear combination of the columns $a_1,...,a_n$ of $A$: so that $b\in\text{col}(A)=\text{range}(A)$ (the column space of $A$).

The relative condition number of a matrix-vector multiply is bounded by relative to a particular matrix norm $\lVert\cdot\rVert$.

In the matrix-matrix multiply $B=AC$, each column of $C$ is multiplied by $A$, so that each column of $B$ is a linear combination of the columns of $A$. Additionally, each row of $B$ is a linear combination of the rows of $C$.