Let's solve this problem step by step.

Part (a): Finding the Change of Basis Matrix $P_{C \leftarrow B}$

We have two bases $B$ and $C$ for $P_3$:

• $B = \{1, t, t^2, t^3\}$
• $C = \{1, t - 1, (t - 1)^2, (t - 1)^3\}$

The change of basis matrix $P_{C \leftarrow B}$ transforms coordinates from basis $B$ to basis $C$. To find this matrix, we need to express each element of the basis $C$ as a linear combination of the basis elements of $B$.

1. Express each element of $C$ in terms of $B$:

   • $1$ is already a part of the basis $B$, so: $1 = 1 \cdot 1 + 0 \cdot t + 0 \cdot t^2 + 0 \cdot t^3$ Coefficients: $(1, 0, 0, 0)$.

   • $t - 1$: $t - 1 = 0 \cdot 1 + 1 \cdot t + 0 \cdot t^2 + 0 \cdot t^3$ Coefficients: $(-1, 1, 0, 0)$.

   • $(t - 1)^2$: $(t - 1)^2 = t^2 - 2t + 1$ Coefficients: $(1, -2, 1, 0)$.

   • $(t - 1)^3$: $(t - 1)^3 = t^3 - 3t^2 + 3t - 1$ Coefficients: $(-1, 3, -3, 1)$.

2. Construct the matrix $P_{C \leftarrow B}$ using these coefficients:

   1 & -1 & 1 & -1 \\ 0 & 1 & -2 & 3 \\ 0 & 0 & 1 & -3 \\ 0 & 0 & 0 & 1 \end{pmatrix}$$

Part (b): Using the Change of Basis Matrix

To write $p(t) = 1 - t + 2t^3$ as a linear combination of the basis vectors in $C$, we need to transform the coordinate vector of $p(t)$ with respect to basis $B$ into the coordinates with respect to basis $C$.

1. Find the coordinate vector of $p(t)$ with respect to $B$: $p(t) = 1 - t + 0 \cdot t^2 + 2 \cdot t^3$ So, the coordinate vector with respect to $B$ is: $[p]_B = \begin{pmatrix} 1 \\ -1 \\ 0 \\ 2 \end{pmatrix}$

2. Find $[p]_C$ using $P_{C \leftarrow B}$: Multiply $P_{C \leftarrow B}$ by $[p]_B$: [ [p]C = P{C \leftarrow B} \cdot [p]_B = \begin{pmatrix} 1 & -1 & 1 & -1 \ 0 & 1 & -2 & 3 \ 0 & 0 & 1 & -3 \ 0 & 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} 1 \ -1 \ 0 \ 2 \end{pmatrix} ]

3. Perform the matrix multiplication:

   1(-1) - 1(-1) + 1(0) - 1(2) = -2 \\ 0(1) + 1(-1) - 2(0) + 3(2) = 5 \\ 0(0) + 0(1) + 1(0) - 3(2) = -6 \\ 0(0) + 0(0) + 0(0) + 1(2) = 2 \end{pmatrix} = \begin{pmatrix} -2 \\ 5 \\ -6 \\ 2 \end{pmatrix}$$

So, the coordinate vector of $p(t)$ with respect to basis $C$ is $[p]_C = (-2, 5, -6, 2)$.

Final Answers:

• (a) The change of basis matrix $P_{C \leftarrow B}$ is: 1 & -1 & 1 & -1 \\ 0 & 1 & -2 & 3 \\ 0 & 0 & 1 & -3 \\ 0 & 0 & 0 & 1 \end{pmatrix}$$
• (b) The polynomial $p(t) = 1 - t + 2t^3$ can be written as: $p(t) = -2 \cdot 1 + 5 \cdot (t - 1) - 6 \cdot (t - 1)^2 + 2 \cdot (t - 1)^3$

Would you like more details or have any questions?

Related Questions:

1. How do you find the change of basis matrix in general for any vector spaces?
2. Why is it important to understand change of basis matrices in linear algebra?
3. How does a change of basis affect the geometric representation of vectors?
4. What is the inverse of a change of basis matrix, and how do you compute it?
5. How can the concept of change of basis be applied to computer graphics and 3D modeling?

Tip:

When working with change of basis matrices, it's crucial to keep track of the order of basis vectors and ensure that matrix multiplication respects this order to avoid errors.