Find the standard matrix for a linear transformation

If T: $\Bbb R$3→ $\Bbb R$3 is a linear transformation such that:

$$ T \Bigg (\begin{bmatrix}-2 \\ 3 \\ -4 \\ \end{bmatrix} \Bigg) = \begin{bmatrix} 5\\ 3 \\ 14 \\ \end{bmatrix}$$ $$T \Bigg (\begin{bmatrix} 3 \\ -2 \\ 3 \\ \end{bmatrix} \Bigg) = \begin{bmatrix}-4 \\ 6 \\ -14 \\ \end{bmatrix}$$ $$ T\Bigg (\begin{bmatrix}-4 \\ -5 \\ 5 \\ \end{bmatrix} \Bigg) = \begin{bmatrix} -6\\ -40 \\ -2 \\ \end{bmatrix} $$

Then the standard matrix for T is...

I'm not exactly sure how to approach this problem. Could anyone explain how to solve this problem?

Solution 1:

The standard matrix has columns that are the images of the vectors of the standard basis $$ T \Bigg (\begin{bmatrix}1\\0\\0\end{bmatrix} \Bigg), \qquad T \Bigg (\begin{bmatrix} 0\\1\\0 \end{bmatrix} \Bigg), \qquad T\Bigg (\begin{bmatrix}0\\0\\1 \end{bmatrix}\Bigg). \tag{1} $$ So one approach would be to solve a system of linear equations to write the vectors of the standard basis in terms of your vectors $$ \begin{bmatrix}-2 \\ 3 \\ -4 \\ \end{bmatrix}, \qquad \begin{bmatrix} 3 \\ -2 \\ 3 \\ \end{bmatrix}, \qquad \begin{bmatrix}-4 \\ -5 \\ 5 \\ \end{bmatrix}, $$ and then obtain (1).

Alternatively, note that if $A$ is the standard matrix you are looking for, then $$ A \cdot \begin{bmatrix}-2 & 3 & -4\\ 3 &-2&-5 \\ -4&3&5 \\ \end{bmatrix} = \begin{bmatrix} 5 & -4 & -6\\ 3 & 6 & -40 \\ 14 & -14 & -2 \\ \end{bmatrix}, $$ and multiply on the right by the inverse of $$ \begin{bmatrix}-2 & 3 & -4\\ 3 &-2&-5 \\ -4&3&5 \\ \end{bmatrix}. $$

Spoiler And the matrix $A$ is...

$$\begin{bmatrix}-1& 5& 3\\ 5& 3&-1\\-2& -2 & -4\end{bmatrix}$$

Many Thanks to @MartinSleziak for correcting two misprints in comments below.

Solution 2:

You can put into a matrix given vectors and their images. If you then do elementary row operations, this property is not changed. (After each step you have in each row a vector and its image. This is because of linearity.) If you manage to obtain the identity matrix on the left, then you know the images of the vectors from the standard basis, which is sufficient to obtain the matrix of your linear transformation. (Since you're using column vectors, the result is the transpose of the matrix on the right. It is worth mentioning that some authors prefer row vectors - in such case the matrix wouldn't been transposed. See also the Wikipedia article on Row and column vectors.)

$ \left(\begin{array}{ccc|ccc} -2 & 3 & -4 & 5 & 3 & 14\\ 3 & -2 & 3 & -4 & 6 &-14\\ -4 & -5 & 5 & -6 &-40&-2 \end{array}\right)\sim \left(\begin{array}{ccc|ccc} 1 & 1 & -1 & 1 & 9 & 0\\ 3 & -2 & 3 & -4 & 6 &-14\\ -4 & -5 & 5 & -6 &-40&-2 \end{array}\right)\sim \left(\begin{array}{ccc|ccc} 1 & 1 & -1 & 1 & 9 & 0\\ 0 & -5 & 6 & -7 &-21 &-14\\ -4 & -5 & 5 & -6 &-40&-2 \end{array}\right)\sim \left(\begin{array}{ccc|ccc} 1 & 1 & -1 & 1 & 9 & 0\\ 0 & -5 & 6 & -7 &-21 &-14\\ 0 & -1 & 1 & -2 & -4 &-2 \end{array}\right)\sim \left(\begin{array}{ccc|ccc} 1 & 0 & 0 & -1 & 5 &-2\\ 0 & -5 & 6 & -7 &-21 &-14\\ 0 & -1 & 1 & -2 & -4 &-2 \end{array}\right)\sim \left(\begin{array}{ccc|ccc} 1 & 0 & 0 & -1 & 5 &-2\\ 0 & 1 & -1 & 2 & 4 & 2\\ 0 & -5 & 6 & -7 &-21 &-14 \end{array}\right)\sim \left(\begin{array}{ccc|ccc} 1 & 0 & 0 & -1 & 5 &-2\\ 0 & 1 & -1 & 2 & 4 & 2\\ 0 & 0 & 1 & 3 & -1 &-4 \end{array}\right)\sim \left(\begin{array}{ccc|ccc} 1 & 0 & 0 & -1 & 5 &-2\\ 0 & 1 & 0 & 5 & 3 &-2\\ 0 & 0 & 1 & 3 & -1 &-4 \end{array}\right) $

Solution 3:

Note that if $$ \begin{pmatrix}-2 \\ 3 \\ -4 \\ \end{pmatrix} = -2\begin{pmatrix}1 \\ 0\\ 0 \\ \end{pmatrix} +3\begin{pmatrix}0 \\ 1\\ 0 \\ \end{pmatrix} -4\begin{pmatrix}0 \\ 0\\ 1 \\ \end{pmatrix} $$ so $$ T\begin{pmatrix}-2 \\ 3 \\ -4 \\ \end{pmatrix} = -2\times T\begin{pmatrix}1 \\ 0\\ 0 \\ \end{pmatrix} +3\times T\begin{pmatrix}0 \\ 1\\ 0 \\ \end{pmatrix} -4\times T\begin{pmatrix}0 \\ 0\\ 1 \\ \end{pmatrix} =-2T(\epsilon_1)+3T(\epsilon_2)-4T(\epsilon_3)$$ Now do the same for two other vectors to find out two relations written by to $T(\epsilon_i)$. Here, you have a system of 3 equations and 3 unknowns $T(\epsilon_i)$ which by solving that you get $T(\epsilon_i)_1^3$. Now use that fact that $$T\begin{pmatrix}x \\y \\ z \\ \end{pmatrix} = xT(\epsilon_1)+yT(\epsilon_2)+zT(\epsilon_3)$$ to find the original relation for $T$. I think by its rule you can find the associated matrix.