Multiplying a matrix by a vector is not a new idea. The vector supplies one weight per column, and the answer is the linear combination of the columns with those weights.

A shopping list gives a quantity for each product. The bill is those quantities applied to the price columns and added up, one product at a time.

For an $m\times n$ matrix $A$ and a vector $\mathbf{x}$ of $\mathbb{R}^n$, the product $A\mathbf{x}$ weights column j by $x_j$ and adds. The vector must have one component per column, and the answer lives in $\mathbb{R}^m$, where the columns live. This is the vector equation of a system, from the lesson on systems, written in one symbol.

• $A\mathbf{x}$ is defined only when $\mathbf{x}$ has as many components as $A$ has columns.
• $A\mathbf{x}$ lands in $\mathbb{R}^m$: it is a combination of vectors that live there.

The columns of $A$ are $\mathbf{a}_1$ and $\mathbf{a}_2$, and $\mathbf{x}$ supplies one weight for each. With weights 2 and 1: two of the first column, then one of the second, and the green arrow is . The matrix contributes the arrows, the vector contributes the scalars.

Unfolding $A\mathbf{x}$ turns the equation back into the vector equation of a system, whose solvability was already shown to be membership of the span of the columns. The matrix notation adds no mathematics; it names the same statement more briefly.

Unfold the product using its definition, which is the only thing the symbol means.

This is exactly the vector equation a system of linear equations was shown to be.

That equation has a solution precisely when $\mathbf{b}$ lies in the span of the columns, as the lesson on systems proved.

The matrix form is a shorter name for the same statement, so it inherits the same answer.

Four of the first column and five of the second, added: $(4,23)$.

$A$ is $3\times2$, $\mathbf{x}\in\mathbb{R}^2$, $A\mathbf{x}\in\mathbb{R}^3$

Two columns need two weights, and each column has three entries.

One of the second column and none of any other.

The zero on the right has as many components as $A$ has rows.