To find the matrix $[T]_E$ in the standard basis $E$ of $\mathbb{R}^2$, given that we have a transformation $T$ with respect to the basis $C = \{\vec{c}_1, \vec{c}_2\}$, we can follow these steps:

1. Write down the transformation matrices for each vector in basis $C$:

   • We have $[T(\vec{c}_1)]_C = \begin{pmatrix} 1 \\ 0 \end{pmatrix}$ and $[T(\vec{c}_2)]_C = \begin{pmatrix} 4 \\ 2 \end{pmatrix}$.

2. Interpret the information as coordinates in basis $C$:

   • This tells us that $T(\vec{c}_1) = 1 \cdot \vec{c}_1 + 0 \cdot \vec{c}_2 = \vec{c}_1$.
   • Similarly, $T(\vec{c}_2) = 4 \cdot \vec{c}_1 + 2 \cdot \vec{c}_2$.

3. Express $T(\vec{c}_1)$ and $T(\vec{c}_2)$ in terms of the standard basis $E$:

   • $\vec{c}_1 = \begin{pmatrix} 2 \\ 3 \end{pmatrix}$
   • $\vec{c}_2 = \begin{pmatrix} 3 \\ 5 \end{pmatrix}$

   Therefore:

   • $T(\vec{c}_1) = 1 \cdot \begin{pmatrix} 2 \\ 3 \end{pmatrix} = \begin{pmatrix} 2 \\ 3 \end{pmatrix}$
   • $T(\vec{c}_2) = 4 \cdot \begin{pmatrix} 2 \\ 3 \end{pmatrix} + 2 \cdot \begin{pmatrix} 3 \\ 5 \end{pmatrix} = \begin{pmatrix} 8 + 6 \\ 12 + 10 \end{pmatrix} = \begin{pmatrix} 14 \\ 22 \end{pmatrix}$

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

   • The matrix $[T]_E$ has columns given by $T(\vec{c}_1)$ and $T(\vec{c}_2)$ expressed in the standard basis $E$: $[T]_E = \begin{pmatrix} 2 & 14 \\ 3 & 22 \end{pmatrix}$

Thus, the matrix representation of $T$ in the standard basis $E$ is: $[T]_E = \begin{pmatrix} 2 & 14 \\ 3 & 22 \end{pmatrix}$

Would you like further details on any of the steps?

Here are some related questions to explore:

1. How can we confirm that a given transformation matrix represents a linear transformation?
2. What are the properties of basis transformations in linear algebra?
3. How does changing the basis affect the matrix representation of a transformation?
4. How can we calculate the inverse of a transformation matrix?
5. What does it mean for a matrix to be similar to another in the context of basis transformations?

Tip: Always ensure you have the correct basis vectors when performing basis transformations, as mistakes in the basis can lead to incorrect results.