Math Problem Statement

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 .

Solution

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

Part (a): Show that TT is a linear transformation.

To show that TT 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)T(u + v) = T(u) + T(v) for all u,vP3u, v \in P_3.
  2. T(cu)=cT(u)T(cu) = cT(u) for all uP3u \in P_3 and cRc \in \mathbb{R}.

Let's consider the transformation T:P3R4T: P_3 \rightarrow \mathbb{R}^4 defined by:

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.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Linear Transformations
Polynomial Functions
Matrix Representation

Formulas

T(p) = [p(-3), p(-1), p(1), p(3)]
Matrix representation of T for the basis {1, t, t^2, t^3}

Theorems

Linear transformation properties (additivity and homogeneity)
Basis and matrix representation

Suitable Grade Level

College Level - Linear Algebra