To find the matrix $[T]_E$ of the linear transformation $T$ with respect to the standard basis $E$ in $\mathbb{R}^2$, we can use the following approach:

Given:

• Basis $C = \{ \vec{c}_1, \vec{c}_2 \}$ with $\vec{c}_1 = \begin{pmatrix} 2 \\ -3 \end{pmatrix}$ and $\vec{c}_2 = \begin{pmatrix} -3 \\ 5 \end{pmatrix}$.
• $[T(\vec{c}_1)]_C = \begin{pmatrix} 1 \\ 0 \end{pmatrix}$ and $[T(\vec{c}_2)]_C = \begin{pmatrix} 4 \\ 2 \end{pmatrix}$.

Steps:

1. Interpret the Action of $T$ in Basis $C$: The coordinates $[T(\vec{c}_1)]_C = \begin{pmatrix} 1 \\ 0 \end{pmatrix}$ and $[T(\vec{c}_2)]_C = \begin{pmatrix} 4 \\ 2 \end{pmatrix}$ tell us that: $T(\vec{c}_1) = 1 \cdot \vec{c}_1 + 0 \cdot \vec{c}_2 = \vec{c}_1 = \begin{pmatrix} 2 \\ -3 \end{pmatrix}$ $T(\vec{c}_2) = 4 \cdot \vec{c}_1 + 2 \cdot \vec{c}_2 = 4 \begin{pmatrix} 2 \\ -3 \end{pmatrix} + 2 \begin{pmatrix} -3 \\ 5 \end{pmatrix}$

2. Compute $T(\vec{c}_2)$ in Standard Basis: Calculate each component of $T(\vec{c}_2)$ separately: $T(\vec{c}_2) = 4 \begin{pmatrix} 2 \\ -3 \end{pmatrix} + 2 \begin{pmatrix} -3 \\ 5 \end{pmatrix} = \begin{pmatrix} 8 \\ -12 \end{pmatrix} + \begin{pmatrix} -6 \\ 10 \end{pmatrix} = \begin{pmatrix} 2 \\ -2 \end{pmatrix}$

3. Construct the Matrix $[T]_E$: The columns of $[T]_E$ are given by $T(\vec{c}_1)$ and $T(\vec{c}_2)$ expressed in the standard basis. Therefore: $[T]_E = \begin{pmatrix} 2 & 2 \\ -3 & -2 \end{pmatrix}$

Final Answer:

$[T]_E = \begin{pmatrix} 2 & 2 \\ -3 & -2 \end{pmatrix}$

Would you like further clarification on any step, or have questions?

Here are five related questions:

1. How would you verify if $T$ is invertible using $[T]_E$?
2. How does the matrix change if the basis $C$ is altered?
3. What is the significance of the standard basis $E$ in this transformation?
4. How can we interpret $[T(\vec{c}_1)]_C$ and $[T(\vec{c}_2)]_C$ geometrically?
5. How does the choice of basis affect the computation of $[T]_E$?

Tip: In linear transformations, the columns of the transformation matrix often reveal how the basis vectors are transformed, helping us understand the transformation's effect on the entire space.