Matrix Product (Conventional)/Examples/Column Matrix All 0 except for One 1

Example of (Conventional) Matrix Product

Let $\mathbf A$ be a matrix of order $m \times n$.

For $1 \le i \le n$, let $\mathbf e_i$ be the column matrix of order $n \times 1$ defined as:

$e_k = \delta_{k i}$

where:

$e_k$ is the element of $\mathbf e_i$ whose indices are $\tuple {k, 1}$

$\delta_{k i}$ denotes the Kronecker delta.

Then $\mathbf A \mathbf e_i$ is the column matrix which is equal to the $i$th column of $\mathbf A$.

Let $\mathbf B = \mathbf A \mathbf e_i$

By definition of matrix product, $\mathbf B$ is a matrix of order $m \times 1$.

Let $b_j$ denote the element of $\mathbf B$ whose indices are $\tuple {j, 1}$.

We have:

$\ds b_j$

$=$

$\ds \sum_{k \mathop = 1}^n a_{j k} e_k$

$\ds $

$=$

$\ds \sum_{k \mathop = 1}^n a_{j k} \delta_{k i}$

Definition of $\mathbf e_i$

$\ds $

$=$

$\ds a_{j i}$

Hence the result.

$\blacksquare$

1998: Richard Kaye and Robert Wilson: Linear Algebra ... (previous) ... (next): Part $\text I$: Matrices and vector spaces: $1$ Matrices: Exercises: $1.9$