To find the matrix $[T]_E$, which represents the linear transformation $T$ in the standard basis $E$ for $\mathbb{R}^2$, we can use the information provided about how $T$ acts on the basis vectors $\vec{c}_1$ and $\vec{c}_2$ of the basis $C$.

Here are the steps:

1. Express $T(\vec{c}_1)$ and $T(\vec{c}_2)$ in terms of the $C$-basis vectors:

   • We are given: $[T(\vec{c}_1)]_C = \begin{pmatrix}1 \\ 0\end{pmatrix}$ This means that $T(\vec{c}_1) = 1 \cdot \vec{c}_1 + 0 \cdot \vec{c}_2 = \vec{c}_1$.

   • Similarly, we have: $[T(\vec{c}_2)]_C = \begin{pmatrix}4 \\ 2\end{pmatrix}$ This means that $T(\vec{c}_2) = 4 \cdot \vec{c}_1 + 2 \cdot \vec{c}_2$.

2. Convert $T(\vec{c}_1)$ and $T(\vec{c}_2)$ to standard basis coordinates:

   • Since $\vec{c}_1 = \begin{pmatrix}2 \\ -3\end{pmatrix}$ and $\vec{c}_2 = \begin{pmatrix}-3 \\ 5\end{pmatrix}$, we calculate: $T(\vec{c}_1) = \vec{c}_1 = \begin{pmatrix}2 \\ -3\end{pmatrix}$ and $T(\vec{c}_2) = 4 \cdot \vec{c}_1 + 2 \cdot \vec{c}_2 = 4 \begin{pmatrix}2 \\ -3\end{pmatrix} + 2 \begin{pmatrix}-3 \\ 5\end{pmatrix}$ $= \begin{pmatrix}8 \\ -12\end{pmatrix} + \begin{pmatrix}-6 \\ 10\end{pmatrix} = \begin{pmatrix}2 \\ -2\end{pmatrix}$

3. Construct the matrix $[T]_E$:

   • The matrix $[T]_E$ is formed by placing the coordinates of $T(\vec{c}_1)$ and $T(\vec{c}_2)$ as columns: $[T]_E = \begin{pmatrix}2 & 2 \\ -3 & -2\end{pmatrix}$

Thus, the matrix representation of $T$ in the standard basis $E$ is: $[T]_E = \begin{pmatrix}2 & 2 \\ -3 & -2\end{pmatrix}$

Would you like further explanation on any step?

Here are five questions that build upon this topic:

1. How would the transformation matrix change if we had a different basis for $\mathbb{R}^2$?
2. How do we determine if the transformation $T$ is invertible from its matrix representation?
3. What is the significance of the eigenvalues and eigenvectors of the matrix $[T]_E$?
4. How would we find the transformation matrix in a rotated basis?
5. How can we verify if $T$ preserves the orientation of the basis vectors?

Tip: Always check the linear independence of basis vectors when working with transformations in different bases.