The question in the image involves finding the change of basis matrix from basis $B$ to basis $C$ in the vector space $P_2$, which consists of polynomials of degree 2 or less.

Explanation:

Step 1: Express each vector in $B$ in terms of $C$.

The bases are given as:

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

Each vector in $B$ must be written as a linear combination of the vectors in $C$: $b_i = c_1 \cdot \text{coeff}_1 + c_2 \cdot \text{coeff}_2 + c_3 \cdot \text{coeff}_3$ where $b_i \in B$ and $c_1, c_2, c_3 \in C$.

Step 2: Solve the system for each $b_i$.

For example, for $b_1 = 2 - x + x^2$, solve: $2 - x + x^2 = a(2 + x - x^2) + b(-2 + x^2) + c(3 + x - 2x^2)$ Equating coefficients of $1, x,$ and $x^2$ yields a system of equations. Repeat for the other vectors in $B$.

Step 3: Form the Change of Basis Matrix.

The resulting matrix will be formed by the coefficients obtained, where: