To find the matrix $[T]_E$ of the linear transformation $T$ in the standard basis $E = \{e_1, e_2\}$ of $\mathbb{R}^2$, we need to perform the following steps:

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Step 1: Understand the problem and given data

• The basis $C = \{\mathbf{c}_1, \mathbf{c}_2\}$ is given as: $\mathbf{c}_1 = \begin{bmatrix} 2 \\ 3 \end{bmatrix}, \quad \mathbf{c}_2 = \begin{bmatrix} 3 \\ 5 \end{bmatrix}.$
• The transformation $T$ is defined in terms of its action on the $C$-basis vectors: [ [T(\mathbf{c}_1)]_C = \begin{bmatrix} 1 \ 0 \end{bmatrix}, \quad [T(\mathbf{c}_2)]_C = \begin{bmatrix} 4 \ 2 \end{bmatrix}. ] These are the coordinates of $T(\mathbf{c}_1)$ and $T(\mathbf{c}_2)$ in the basis $C$.

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Step 2: Express $T(\mathbf{c}_1)$ and $T(\mathbf{c}_2)$ in terms of standard basis $E$

To convert the coordinates in basis $C$ into standard basis $E$, use the relation: $T(\mathbf{c}_i) = \sum_{j=1}^2 \text{(coordinate of } T(\mathbf{c}_i) \text{ in basis } C) \cdot \mathbf{c}_j.$

Compute $T(\mathbf{c}_1)$:

$T(\mathbf{c}_1) = 1 \cdot \mathbf{c}_1 + 0 \cdot \mathbf{c}_2 = \mathbf{c}_1 = \begin{bmatrix} 2 \\ 3 \end{bmatrix}.$

Compute $T(\mathbf{c}_2)$:

$T(\mathbf{c}_2) = 4 \cdot \mathbf{c}_1 + 2 \cdot \mathbf{c}_2 = 4 \begin{bmatrix} 2 \\ 3 \end{bmatrix} + 2 \begin{bmatrix} 3 \\ 5 \end{bmatrix}.$ Simplify: $T(\mathbf{c}_2) = \begin{bmatrix} 8 \\ 12 \end{bmatrix} + \begin{bmatrix} 6 \\ 10 \end{bmatrix} = \begin{bmatrix} 14 \\ 22 \end{bmatrix}.$

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Step 3: Form the transformation matrix $[T]_E$

The columns of $[T]_E$ are the images of the standard basis vectors $\mathbf{e}_1 = \begin{bmatrix} 1 \\ 0 \end{bmatrix}$ and $\mathbf{e}_2 = \begin{bmatrix} 0 \\ 1 \end{bmatrix}$, expressed in the standard basis. These are just $T(\mathbf{c}_1)$ and $T(\mathbf{c}_2)$, as computed in Step 2.

Thus: $[T]_E = \begin{bmatrix} T(\mathbf{c}_1) & T(\mathbf{c}_2) \end{bmatrix} = \begin{bmatrix} 2 & 14 \\ 3 & 22 \end{bmatrix}.$

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Final Answer:

$[T]_E = \begin{bmatrix} 2 & 14 \\ 3 & 22 \end{bmatrix}.$

Would you like further explanation or verification of these calculations? Here are some related questions:

1. What are the properties of a linear transformation represented by a matrix?
2. How do changes in basis affect the representation of a linear transformation?
3. Can this method be extended to higher-dimensional spaces?
4. What is the significance of the columns of the transformation matrix?
5. How do you compute the inverse of a transformation matrix if it exists?

Tip: Always verify your calculations by applying the transformation matrix to the original basis vectors!