Matrix–vector product

Definition

Let $R$ be a ring, let $A = (a_{ij})$ be an $m \times n$ matrix over $R$, and let $x = (x_1, \dots, x_n)^T \in R^n$ be a column vector. The matrix–vector product $Ax$ is the column vector in $R^m$ with entries $$(Ax)_i = \sum_{j=1}^{n} a_{ij} x_j$$ for $1 \le i \le m$.

Remarks

If $A_1, \dots, A_n$ are the columns of $A$, then $Ax = x_1 A_1 + \cdots + x_n A_n$, a linear combination of the columns. A system of linear equations is the equation $Ax = b$. The map $x \mapsto Ax$ is linear, see multiplication by a matrix as a linear map. For example, $\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \begin{pmatrix} 1 \\ 1 \end{pmatrix} = \begin{pmatrix} 3 \\ 7 \end{pmatrix}$.

Explanations

L1 human

To multiply a matrix by a vector, take each row of the matrix in turn. Multiply its entries by the entries of the vector, position by position, and add the results. Each row gives one entry of the answer.