91Ó°ÊÓ

A linear transformation is a function from one vector space to the other vector space that respects the linear structure of each vector space.

For example,

The defining characteristic of a linear transformation.

$T:V→W$is that, for any vectors x and y in V and scalars a and b of the underlying fields.

T(ax + by) = aT (x) + bT (y)

The fact that the matrix of a linear transformation T with respect to a basis ${\mathrm{B}}_1=\left(\mathrm{f},\mathrm{g}\right)$ is .

$\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]$

Which implies that the first column gives us the coordinates of the base vectors’ images in that same base, hence

role="math" localid="1659432935908" $$\begin{gathered} \mathrm{T}\left(\mathrm{f}\right)=1-\mathrm{f}+3-\mathrm{g} \\ \mathrm{T}\left(\mathrm{g}\right)=2-\mathrm{f}+4-\mathrm{g} \end{gathered}$$

Since, form a matrix T with respect to the basis role="math" localid="1659432871492" ${\mathrm{B}}_2=(\mathrm{g},\mathrm{f})$.

Which observes these relations (with order change)

$$\begin{gathered} \mathrm{T}\left(\mathrm{f}\right)=4-\mathrm{g}+2-\mathrm{g} \\ \mathrm{T}\left(\mathrm{g}\right)=3-\mathrm{g}+1-\mathrm{f} \end{gathered}$$

Therefore, the matrix would be ${\mathrm{T}}_{{\mathrm{B}}_2}=\left[\begin{array}{cc}4& 3\\ 2& 1\end{array}\right]$.

This matrix does not match the one given in the text.

Hence, the statement is false.