Column Transformations

Elementary matrices can be also used for column transformations when they are multiplied from right. We use Mathematica for their demonstrations. Let us first define an arbitrary 3 × 3 matrix:

${\bf A} = \begin{bmatrix} a&b&c \\ d&e&f \\ g&h&i \end{bmatrix} .$

A = {{a, b, c}, {d, e, f}, {g, h, i}}

We define an elementary matrix for switching first two columns:

${\bf E}_1 = \begin{bmatrix} 0&1&0 \\ 1&0&0 \\ 0&0&1 \end{bmatrix} \qquad \Longrightarrow \qquad {\bf A}\, {\bf E}_1 = \begin{bmatrix} b&a&c \\ e&d&f \\ h&g&i \end{bmatrix} .$

F1 = {{0, 1, 0}, {1, 0, 0}, {0, 0, 1}};
A . F

{{b, a, c}, {e, d, f}, {h, g, i}}

If we want to multiply the second column of an n × 3 matrix by 7, we would multiply from right the matrix by

${\bf E}_2 (7) = \begin{bmatrix} 1&0&0 \\ 0&7&0 \\ 0&0&1 \end{bmatrix} ,$

which is the matrix that results when the second column of I₃ is multiplied by 7. We have

${\bf A}\,{\bf E}_2 (7) = \begin{bmatrix} a&7\,b &c \\ d& 7\,e &f \\ g& 7\,h & i \end{bmatrix} .$

F2 = {{1, 0, 0}, {0, 7, 0}, {0, 0, 1}};
A . F2

{{a, 7 b, c}, {d, 7 e, f}, {g, 7 h, i}}

Similarly, if we multiply from right by matrix

${\bf F}_3 = \begin{bmatrix} 1 & 7 & 0 \\ 0&1&0 \\ 0&0&1 \end{bmatrix} ,$

we get

${\bf A}\,{\bf E}_3 (7) = \begin{bmatrix} a& 7 a + b& c \\ d& 7 d + e& f \\ g& 7 g + h& i \end{bmatrix} .$

F3 = {{1, 7, 0}, {0, 1, 0}, {0, 0, 1}};
A . F3

{{a, 7 a + b, c}, {d, 7 d + e, f}, {g, 7 g + h, i}}

Axier, S., Linear Algebra Done Right. Undergraduate Texts in Mathematics (3rd ed.). Springer. 2015, ISBN 978-3-319-11079-0.
Beezer, R.A., A First Course in Linear Algebra, 2017.
Dillon, M., Linear Algebra, Vector Spaces, and Linear Transformations, American Mathematical Society, Providence, RI, 2023.

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Column Transformations