Matrix-Vector Product

$$\begin{array}{r}A=[{\vec{a}}_1{\vec{a}}_2...{\vec{a}}_n]\in M_{m\times n}(\mathbb{R})\\ \\ \vec{x}=\left[\begin{array}{c}{x}_1\\ .\\ .\\ x_n\end{array}\right]\in {\mathbb{R}}^n\\ \\ A\vec{x}=x_1{\vec{a}}_1+...+x_n{\vec{a}}_n\in R^m\end{array}$$

• We have a matrix which is a bunch of vectors together, where each vector is a column of the matrix.
• Then we have the solution vector x which is a bunch of xs for each dimension collected together.
• The matrix-vector product is the sum of products between the corresponding column vector and x in that dimension.
• $A\vec{x}$ is a linear combination of elements of $\vec{x}$ and columns of $A$.

Alternative way to think of this

Row dot Column.

• The $nth$ row of the resultant matrix is a dot product between the $nth$ row of the matrix and the vector $\vec{x}$.
• Has just one column as dot product of vector sums up the product of the corresponding entries.