Finding If A linear Transformation Exists

Question

Let $T:\mathbb{R}^3 \to \mathbb{R}^3$ defined as followed: $$T\begin{pmatrix} 3 \\ 2\\ 5 \end{pmatrix}=\begin{pmatrix} 1 \\ 3\\ 2 \end{pmatrix},T\begin{pmatrix} 2 \\ 2\\ 5 \end{pmatrix}=\begin{pmatrix} 0 \\ 1\\ 0 \end{pmatrix},T\begin{pmatrix} 1 \\ 1\\ 3 \end{pmatrix}=\begin{pmatrix} 0 \\ 1\\ 1 \end{pmatrix}$$

If there such a linear transformation find it

What I have done is to to set $$T= \begin{pmatrix} a & b & c\\ d & e & f\\ h & k & l \end{pmatrix}$$

And create linear equations to find that

$$T= \begin{pmatrix} 1 & -1 & 0\\ 2 & -4 & 1\\ 2 & -7 & 2 \end{pmatrix}$$

But $\begin{pmatrix} 1 \\ 3\\ 2 \end{pmatrix},\begin{pmatrix} 0 \\ 1\\ 0 \end{pmatrix},\begin{pmatrix} 0 \\ 1\\ 1 \end{pmatrix}$ is a basis for $\mathbb{R}^3$, and by knowing how the linear transformation work on a basis we can find how it work on any vector, but how can it be done?

Answer 1

$u_1 = \pmatrix{3\\2\\5},u_2 = \pmatrix{2\\2\\5}, u_3=\pmatrix{1\\1\\3}\\ u_1 - u_2 = \pmatrix{1\\0\\0}$

$T\pmatrix{1\\0\\0} = Tu_1 - Tu_2$

Now find $T\pmatrix{0\\1\\0}$ and $T\pmatrix{0\\0\\1}$ and you will have your matrix (in the standard basis.)

Now it would be an option to make $u_1,u_2, u_3$ the basis then you would have to find what $Tu_1,Tu_2, Tu_3$ looks like in this basis.

Or you could make $\begin{pmatrix}1 \\3\\2\end{pmatrix},\begin{pmatrix}0 \\ 1\\0\end{pmatrix},\begin{pmatrix} 0 \\1\\1\end{pmatrix}$ your basis as you suggest but then you will need to find what $u_1, u_2, u_3$ look like in this basis.

Answer 2

You have 9 variables in your matrix $T$ and if you multiply $T$ by the vectors in your definition then you will find that you have 9 equations. With a little algebra you can find all 9 elements of $T$.

Answer 3

Guide:

Verify if $\begin{bmatrix} v_1, v_2, v_3\end{bmatrix}$ is non-singular (or check if $\{ v_1, v_2, v_3\}$ is linearly independent). If it is then such linear transfomation $T$ exists. Given any vector $x$, we can express them as $x=\sum_{i=1}^3 \alpha_iv_i$ and hence $T(x)=\sum_{i=1}^3 \alpha_iT(v_i)$. In particular, you can let $x$ be the standard unit vector.

If it is singular, then you have to check for consistency, let say we found that $v_3=\sum_{i=1}^2 \beta_i v_i$, then you have to verify that $T(v_3)=\sum_{i=1}^2 \beta_i T(v_i)$ for such linear transformation to exists.