To solve this problem, we’ll start by constructing the matrix of the transformation $T$ relative to the basis $B$ (part (a)), and then use that to find the matrix of $T$ relative to the standard basis $E$ (part (b)).

Part (a): Matrix of $T$ relative to basis $B$

Given:

• $b_1 = \begin{bmatrix} 1 \\ 1 \end{bmatrix}$
• $b_2 = \begin{bmatrix} 1 \\ 0 \end{bmatrix}$

The transformation $T$ is defined by:

• $T(b_1) = 8b_1 + 6b_2$
• $T(b_2) = 7b_1 + 4b_2$

In terms of the basis $B = \{b_1, b_2\}$, the images of $b_1$ and $b_2$ can be expressed as coordinate vectors relative to $B$:

• $T(b_1) = 8b_1 + 6b_2$ corresponds to the vector $\begin{bmatrix} 8 \\ 6 \end{bmatrix}$ in the $B$-basis.
• $T(b_2) = 7b_1 + 4b_2$ corresponds to the vector $\begin{bmatrix} 7 \\ 4 \end{bmatrix}$ in the $B$-basis.

Thus, the matrix of $T$ relative to the basis $B$ is: $[T]_B = \begin{bmatrix} 8 & 7 \\ 6 & 4 \end{bmatrix}$

Part (b): Matrix of $T$ relative to the standard basis $E$

To find the matrix of $T$ relative to the standard basis $E$, we need to express $T(b_1)$ and $T(b_2)$ in terms of the standard basis vectors.

1. Compute $T(b_1)$ and $T(b_2)$ in terms of the standard basis:

   • Since $b_1 = \begin{bmatrix} 1 \\ 1 \end{bmatrix}$ and $b_2 = \begin{bmatrix} 1 \\ 0 \end{bmatrix}$,
   • $T(b_1) = 8b_1 + 6b_2 = 8 \begin{bmatrix} 1 \\ 1 \end{bmatrix} + 6 \begin{bmatrix} 1 \\ 0 \end{bmatrix} = \begin{bmatrix} 8 + 6 \\ 8 \end{bmatrix} = \begin{bmatrix} 14 \\ 8 \end{bmatrix}$,
   • $T(b_2) = 7b_1 + 4b_2 = 7 \begin{bmatrix} 1 \\ 1 \end{bmatrix} + 4 \begin{bmatrix} 1 \\ 0 \end{bmatrix} = \begin{bmatrix} 7 + 4 \\ 7 \end{bmatrix} = \begin{bmatrix} 11 \\ 7 \end{bmatrix}$.

2. Construct $[T]_E$: Using these results, we have $[T]_E = \begin{bmatrix} 14 & 11 \\ 8 & 7 \end{bmatrix}$

Final Answer

• (a) The matrix of $T$ relative to basis $B$ is: $[T]_B = \begin{bmatrix} 8 & 7 \\ 6 & 4 \end{bmatrix}$

• (b) The matrix of $T$ relative to the standard basis $E$ is: $[T]_E = \begin{bmatrix} 14 & 11 \\ 8 & 7 \end{bmatrix}$

Would you like a more detailed breakdown or have any questions?

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Here are five related questions to deepen understanding:

1. How would the transformation matrix $T$ change if $b_1$ and $b_2$ were swapped in the basis $B$?
2. What does the transformation $T$ represent geometrically in $\mathbb{R}^2$?
3. How would you find the eigenvalues and eigenvectors of $T$ with respect to the basis $B$?
4. How would you verify that $B$ is indeed a basis for $\mathbb{R}^2$?
5. What is the determinant of $[T]_E$, and what does it tell you about $T$?

Tip: When changing between different bases, understanding the relationship between basis vectors helps streamline transformation matrix calculations.