Linear Transformations given vector I'm having trouble understanding how to solve linear transformations given a vector transforming into another vector. Any suggestions for the example below? This is not merely someone trying to get a homework answer, but rather a simple explanation to someone taking Linear Algebra for the first time.


  Let $T:\mathbb{R}^2\to \mathbb{R}^2$ be a linear transformation that sends the vector $u=(5,2)$ into $(2,1)$ and maps $v =(1,3)$ into $(−1,3)$. Calculate $T(-3u), T(7v),$ and $T(-3u+7v)$.
Think of the definition of a linear transformation. Suppose $T$ is a linear transformation from $\mathbb{R}^n \mapsto \mathbb{R}^n$. Then $T$ satisfies the followings:

(1) For all $x,y \in \mathbb{R}^n$, $T(x+y) = T(x)+T(y)$ (Addition)

(2) For all $\lambda \in \mathbb{R}$ and $x \in \mathbb{R}^n$, $T(\lambda x) = \lambda T(x)$ (Scalar multiplication)

These are things that we need for solving the problem. In other words, there is no need for calculating the exact value of the matrix representation of $T$. In the problem, we have
$$ T(u) = \begin{pmatrix}2\\ 1\end{pmatrix}, \quad T(v) = \begin{pmatrix}-1 \\ 3\end{pmatrix}$$
for $u^T = (5,2)$ and $v^T = (1,3)$. And the calculation for $T(-3u),T(7v),$ and $T(-3u+7v)$ can be derived from the definition above.
$$\begin{align}T(-3u) &amp;= -3T(u) \quad (\text{by the definition (2)})\\
&amp;= -3\begin{pmatrix}2 \\ 1\end{pmatrix}\\
&amp;= \begin{pmatrix} -6 \\ -3\end{pmatrix}\end{align}$$
Likewise, $T(7v)$ becomes $\begin{pmatrix}-7\\ 21\end{pmatrix}$. By using the definition (1), the calculation of $T(-3u + 7v)$ becomes $T(-3u) + T(7v)$ and we can easily get the result by simply adding up the previous calculations.
You can pose this as a linear least squares problem if you first remake your matrix into a vector with a process called vectorization, then we can write multiplication with another matrix or a vector as a matrix multiplication with a vectorized matrix. You can find a quite good description how to do this on This wikipedia site under "Matrix equations".

You will get a matrix equation system of the type $\bf Mx =b$ which you can solve for example with linear least squares.