I'm having a really hard time understanding the logic behind linear transformations and its applications.
Let $S$ be the square with vertices's $(0,0),\,(1,0),\,(0,1),\,(1,1)$ and $P$ the parallelogram with the vertices's $(0,0),\,(3,1),\,(2,3),\,(5,4)$. Find a matrix $A$ such that the linear transformation of $T_1 :R^2\rightarrow R^2$ defined by $T(x)=Ax$ for every $x \in R^2$ maps the square $S$ onto the parallelogram $P$
What I understand is the following transformations occur
$$
T
\left[
\begin{array}{c}
1\\
0\\
\end{array}
\right]
=\left[
\begin{array}{c}
3\\
1\\
\end{array}
\right],\,\,\,
T
\left[
\begin{array}{c}
0\\
1\\
\end{array}
\right]
=\left[
\begin{array}{c}
2\\
3\\
\end{array}
\right]
$$
How do I find the Matrix $A$ that satifies these transformations? Explain it like I'm five. Thanks!
Best Answer
Hint: Given any $\vec x = (x_1, x_2) \in \mathbb R^2$, notice that: $$ T(\vec x) = T\left(\begin{bmatrix} x_1 \\ x_2 \end{bmatrix} \right) = T\left(x_1\begin{bmatrix} 1 \\ 0 \end{bmatrix} + x_2\begin{bmatrix} 0 \\ 1 \end{bmatrix} \right) = x_1T\left(\begin{bmatrix} 1 \\ 0 \end{bmatrix} \right) + x_2T\left(\begin{bmatrix} 0 \\ 1 \end{bmatrix} \right) $$ Now substitute the two relationships that you wrote out and rewrite the result in terms of a matrix-vector product.