Problem Analysis

The task is to compute the change of basis matrices for the vector space $P_2$, the space of polynomials of degree $\leq 2$. We are given two bases:

1. Basis $B = \{-1 + x - x^2, \, 1 + x^2, \, 1 + x \}$
2. Basis $C = \{-2 - x - x^2, \, -2 - x^2, \, 3 + x + 2x^2\}$

Questions

(a) Find the change of basis matrix from $B$ to $C$, denoted $[\text{id}]_C^B$.

(b) Find the change of basis matrix from $C$ to $B$, denoted $[\text{id}]_B^C$.

Solution Steps

1. Representation of Basis Vectors of $B$ in Terms of $C$

To find $[\text{id}]_C^B$, we express each basis vector of $B$ as a linear combination of the basis vectors of $C$. That is, for $b_1, b_2, b_3 \in B$ and $c_1, c_2, c_3 \in C$: $b_i = a_{i1} c_1 + a_{i2} c_2 + a_{i3} c_3 \quad \text{for } i = 1, 2, 3.$ The coefficients $a_{ij}$ form the columns of $[\text{id}]_C^B$.

2. Solving the System of Equations

For each $b_i$, solve the system of equations resulting from equating $b_i$ to a combination of $c_1, c_2, c_3$. Use the coefficients of $1, x, x^2$ in each polynomial to create and solve the system.

3. Find $[\text{id}]_B^C$

The matrix $[\text{id}]_B^C$ is the inverse of $[\text{id}]_C^B$.

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