Define T W P3 ! R4 by T .p/ D
2
6
6
4
p.￾3/
p.￾1/
p.1/
p.3/
3
7
7
5
.
a. Show that T is a linear transformation.
b. Find the matrix for T relative to the basis f1; t; t 2
; t 3
g for
P3 and the standard basis for R4
.

Let's tackle each part of the problem step-by-step.

### Part (a): Show that \( T \) is a linear transformation.

To show that \( T \) is a linear transformation, we need to verify that it satisfies the properties of additivity and homogeneity:
1. \( T(u + v) = T(u) + T(v) \) for all \( u, v \in P_3 \).
2. \( T(cu) = cT(u) \) for all \( u \in P_3 \) and \( c \in \mathbb{R} \).

Let's consider the transformation \( T: P_3 \rightarrow \mathbb{R}^4 \) defined by:
\[ T(p) = \begin{bmatrix}
p(-3) \\
p(-1) \\
p(1) \\
p(3)
\end{bmatrix} \]

1. **Additivity:**

Let \( p \) and \( q \) be polynomials in \( P_3 \).

\[
T(p + q) = \begin{bmatrix}
(p+q)(-3) \\
(p+q)(-1) \\
(p+q)(1) \\
(p+q)(3)
\end{bmatrix}
= \begin{bmatrix}
p(-3) + q(-3) \\
p(-1) + q(-1) \\
p(1) + q(1) \\
p(3) + q(3)
\end{bmatrix}
= \begin{bmatrix}
p(-3) \\
p(-1) \\
p(1) \\
p(3)
\end{bmatrix}
+ \begin{bmatrix}
q(-3) \\
q(-1) \\
q(1) \\
q(3)
\end{bmatrix}
= T(p) + T(q)
\]

2. **Homogeneity:**

Let \( p \) be a polynomial in \( P_3 \) and \( c \in \mathbb{R} \).

\[
T(cp) = \begin{bmatrix}
(cp)(-3) \\
(cp)(-1) \\
(cp)(1) \\
(cp)(3)
\end{bmatrix}
= \begin{bmatrix}
c \cdot p(-3) \\
c \cdot p(-1) \\
c \cdot p(1) \\
c \cdot p(3)
\end{bmatrix}
= c \cdot \begin{bmatrix}
p(-3) \\
p(-1) \\
p(1) \\
p(3)
\end{bmatrix}
= c \cdot T(p)
\]

Since \( T \) satisfies both additivity and homogeneity, \( T \) is a linear transformation.

### Part (b): Find the matrix for \( T \) relative to the basis \(\{1, t, t^2, t^3\}\) for \( P_3 \) and the standard basis for \( \mathbb{R}^4 \).

We need to find the matrix representation of \( T \) relative to the given bases. Let the basis for \( P_3 \) be \(\{1, t, t^2, t^3\}\).

We need to determine \( T(1) \), \( T(t) \), \( T(t^2) \), and \( T(t^3) \).

1. **\( T(1) \):**

\[ T(1) = \begin{bmatrix}
1(-3) \\
1(-1) \\
1(1) \\
1(3)
\end{bmatrix}
= \begin{bmatrix}
1 \\
1 \\
1 \\
1
\end{bmatrix} \]

2. **\( T(t) \):**

\[ T(t) = \begin{bmatrix}
t(-3) \\
t(-1) \\
t(1) \\
t(3)
\end{bmatrix}
= \begin{bmatrix}
-3 \\
-1 \\
1 \\
3
\end{bmatrix} \]

3. **\( T(t^2) \):**

\[ T(t^2) = \begin{bmatrix}
t^2(-3) \\
t^2(-1) \\
t^2(1) \\
t^2(3)
\end{bmatrix}
= \begin{bmatrix}
9 \\
1 \\
1 \\
9
\end{bmatrix} \]

4. **\( T(t^3) \):**

\[ T(t^3) = \begin{bmatrix}
t^3(-3) \\
t^3(-1) \\
t^3(1) \\
t^3(3)
\end{bmatrix}
= \begin{bmatrix}
-27 \\
-1 \\
1 \\
27
\end{bmatrix} \]

The matrix for \( T \) relative to the basis \(\{1, t, t^2, t^3\}\) for \( P_3 \) and the standard basis for \( \mathbb{R}^4 \) is formed by taking these vectors as columns:

\[
[T] = \begin{bmatrix}
1 & -3 & 9 & -27 \\
1 & -1 & 1 & -1 \\
1 & 1 & 1 & 1 \\
1 & 3 & 9 & 27
\end{bmatrix}
\]

Thus, the matrix for \( T \) is:

\[
[T] = \begin{bmatrix}
1 & -3 & 9 & -27 \\
1 & -1 & 1 & -1 \\
1 & 1 & 1 & 1 \\
1 & 3 & 9 & 27
\end{bmatrix}
\]

Would you like further details on any part of this problem or have any questions? Here are some related questions to expand your understanding:

1. What are the properties of a linear transformation in general?
2. How do you determine the matrix of a linear transformation with respect to different bases?
3. Can you find the inverse of the matrix \( [T] \) if it exists?
4. What is the geometric interpretation of the transformation \( T \) in this context?
5. How does the choice of basis affect the representation of a linear transformation?

**Tip:** When finding the matrix representation of a linear transformation, always apply the transformation to each basis vector and use the results as the columns of the matrix.

Define T: P3 → R4 by T(p) = [p(-3), p(-1), p(1), p(3)].
a. Show that T is a linear transformation.
b. Find the matrix for T relative to the basis {1, t, t^2, t^3} for P3 and the standard basis for R4.

Explore the linear transformation T: P3 → R4 defined by T(p) = [p(-3), p(-1), p(1), p(3)], showing it is a linear transformation. This solution covers matrix representation of T relative to the basis {1, t, t^2, t^3} for P3 and the standard basis for R4. Suitable for college-level students studying linear algebra.

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