Let's solve the exercises presented in the image:

1. Change of basis matrix in $\mathbb{R}^2$

We need to find the change of basis matrix from the basis $B_1 = \{ (1,2), (2,1) \}$ to the basis $B_2 = \{ (1,-1), (-1,1) \}.$

The change of basis matrix from $B_1$ to $B_2$ is found by expressing each vector in $B_2$ as a linear combination of the vectors in $B_1$. This results in a matrix that transforms a vector from the $B_1$ coordinate system to the $B_2$ coordinate system.

Let’s denote the change of basis matrix by $P_{B_1 \to B_2}$. To find this matrix, we need to write the vectors of $B_2$ in terms of the vectors of $B_1$.

• Let $(1,-1) = a(1,2) + b(2,1)$. Solve for $a$ and $b$.
• Let $(-1,1) = c(1,2) + d(2,1)$. Solve for $c$ and $d$.

We will solve these systems of equations to find the coefficients, then construct the matrix.

2. Change of basis matrix in $P_2(\mathbb{R})$

We need to find the change of basis matrix from the basis $B_1 = \{ 1, t, t^2 \}$ to the basis $B_2 = \{ 1+t, 2+t, 1+t+t^2 \}.$

Again, we need to express each vector in $B_2$ as a linear combination of the vectors in $B_1$. This results in a matrix that transforms a polynomial from the $B_1$ coordinate system to the $B_2$ coordinate system.

• Let $1+t = a_1 \cdot 1 + a_2 \cdot t + a_3 \cdot t^2$.
• Let $2+t = b_1 \cdot 1 + b_2 \cdot t + b_3 \cdot t^2$.
• Let $1+t+t^2 = c_1 \cdot 1 + c_2 \cdot t + c_3 \cdot t^2$.

We will solve these systems of equations to find the coefficients and construct the change of basis matrix.

Now, I will calculate the solutions step by step for both parts. Let’s start with the first exercise.For Exercise 1, the change of basis matrix from $B_1 = \{(1,2), (2,1)\}$ to $B_2 = \{(1,-1), (-1,1)\}$ is:

$P_{B_1 \to B_2} = \begin{pmatrix} -1 & 1 \\ 1 & -1 \end{pmatrix}$

Now let's move on to Exercise 2 and calculate the change of basis matrix for polynomials.It seems like I can’t do more advanced data analysis right now. Please try again later. Would you like me to guide you through solving the second exercise step by step manually?

In the meantime, here are some related questions to further your understanding:

1. How do you find a matrix that changes from one basis to another?
2. What does it mean for a matrix to represent a change of basis?
3. How can you verify if a change of basis matrix is correct?
4. What is the geometric interpretation of changing a basis in $\mathbb{R}^2$?
5. How does a change of basis affect the coordinates of a vector?

Tip: When dealing with change of basis, remember that the new matrix transforms coordinates between two systems by expressing each new basis vector as a combination of the old basis vectors.