Let's solve the problem step-by-step.

We have a transformation $T: \mathbb{R}^2 \rightarrow \mathbb{R}^3$ defined by: $T(x, y) = (x + y, x, y)$ and a vector $v = (5, 2)$.

Part (a): Finding the Standard Matrix of $T$

The standard matrix $A$ of $T$ can be found by applying $T$ to the standard basis vectors of $\mathbb{R}^2$: $e_1 = (1, 0)$ and $e_2 = (0, 1)$.

1. Compute $T(e_1)$: $T(1, 0) = (1 + 0, 1, 0) = (1, 1, 0)$

2. Compute $T(e_2)$: $T(0, 1) = (0 + 1, 0, 1) = (1, 0, 1)$

Thus, the columns of the standard matrix $A$ are $T(e_1)$ and $T(e_2)$. So, $A = \begin{bmatrix} 1 & 1 \\ 1 & 0 \\ 0 & 1 \end{bmatrix}$

Part (b): Finding the Matrix Relative to Bases $B$ and $B'$

Given:

• Basis $B = \{ (1, -1), (0, 1) \}$ for $\mathbb{R}^2$
• Basis $B' = \{ (1, 1, 0), (0, 1, 1), (1, 0, 1) \}$ for $\mathbb{R}^3$

We need to find the matrix of $T$ relative to these bases.

To do this, we:

1. Express the images of the basis vectors of $B$ under $T$ (i.e., $T(1, -1)$ and $T(0, 1)$) in terms of the basis $B'$.
2. Then, form the matrix using the coefficients of these vectors in terms of $B'$.

Step 1: Calculate $T(1, -1)$ and $T(0, 1)$

1. For $T(1, -1)$: $T(1, -1) = (1 - 1, 1, -1) = (0, 1, -1)$

2. For $T(0, 1)$: $T(0, 1) = (0 + 1, 0, 1) = (1, 0, 1)$

Step 2: Express $T(1, -1)$ and $T(0, 1)$ in Terms of $B'$

We need to express $(0, 1, -1)$ and $(1, 0, 1)$ as linear combinations of the basis vectors in $B' = \{ (1, 1, 0), (0, 1, 1), (1, 0, 1) \}$.

1. Expressing $(0, 1, -1)$ in terms of $B'$: Find scalars $a, b, c$ such that: $a(1, 1, 0) + b(0, 1, 1) + c(1, 0, 1) = (0, 1, -1)$

2. Expressing $(1, 0, 1)$ in terms of $B'$: Find scalars $d, e, f$ such that: $d(1, 1, 0) + e(0, 1, 1) + f(1, 0, 1) = (1, 0, 1)$

After determining these scalars, we can construct the matrix relative to $B$ and $B'$ using these as columns.

Would you like assistance solving for these scalars, or further details on any step above?

Here are some related questions for further practice:

1. How can we verify the transformation matrix using different bases?
2. What properties change in a linear transformation when switching bases?
3. How do you find the inverse of a matrix with respect to non-standard bases?
4. Why is it important to express vectors in terms of a chosen basis?
5. How do linear transformations affect geometric shapes under different bases?

Tip: When converting vectors to different bases, keep track of each step meticulously, as minor errors can propagate through matrix transformations.