/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 4 Define \(\mathrm{T}: \mathrm{M... [FREE SOLUTION] | 91影视

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Chapter 2: Problem 4

Define \(\mathrm{T}: \mathrm{M}_{2 \times 2}(R) \rightarrow \mathrm{P}_{2}(R) \quad\) by $\quad \mathrm{\top}\left(\begin{array}{ll}a & b \\ c & d\end{array}\right)=(a+b)+(2 d) x+b x^{2} .$ Let $$ \beta=\left\\{\left(\begin{array}{ll} 1 & 0 \\ 0 & 0 \end{array}\right),\left(\begin{array}{ll} 0 & 1 \\ 0 & 0 \end{array}\right),\left(\begin{array}{ll} 0 & 0 \\ 1 & 0 \end{array}\right),\left(\begin{array}{ll} 0 & 0 \\ 0 & 1 \end{array}\right)\right\\} \quad \text { and } \quad \gamma=\left\\{1, x, x^{2}\right\\} $$ Compute \([T]_{\beta}^{\gamma}\).

Short Answer

Expert verified
The matrix representation of the linear transformation T relative to bases 尾 and 纬 is: $$ [T]_\beta^\gamma = \left(\begin{array}{llll} 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 2 \\ 0 & 1 & 0 & 0 \end{array}\right) $$

Step by step solution

01

Apply T to 尾

First, we need to apply the linear transformation T to each basis vector in 尾. Given, $$ T\left(\begin{array}{cc} a & b \\ c & d \end{array}\right) = (a+b) + (2d)x + bx^2 $$ Applying T to each basis vector in 尾, we have: 1. \(T\left(\begin{array}{ll} 1 & 0 \\ 0 & 0 \end{array}\right)\): $$ T\left(\begin{array}{cc} 1 & 0 \\ 0 & 0 \end{array}\right) = (1+0) + (2\cdot 0)x + 0x^2 = 1 $$ 2. \(T\left(\begin{array}{ll} 0 & 1 \\ 0 & 0 \end{array}\right)\): $$ T\left(\begin{array}{cc} 0 & 1 \\ 0 & 0 \end{array}\right) = (0+1) + (2\cdot 0)x + 1x^2 = 1 + x^2 $$ 3. \(T\left(\begin{array}{ll} 0 & 0 \\ 1 & 0 \end{array}\right)\): $$ T\left(\begin{array}{cc} 0 & 0 \\ 1 & 0 \end{array}\right) = (0+0) + (2\cdot 0)x + 0x^2 = 0 $$ 4. \(T\left(\begin{array}{ll} 0 & 0 \\ 0 & 1 \end{array}\right)\): $$ T\left(\begin{array}{cc} 0 & 0 \\ 0 & 1 \end{array}\right) = (0+0) + (2\cdot 1)x + 0x^2 = 2x $$
02

Write as a linear combination of 纬

Let's express the transformed matrix as the linear combination of the basis vectors in 纬, which are {1, x, x虏}. 1. \(T\left(\begin{array}{ll} 1 & 0 \\ 0 & 0 \end{array}\right) = 1 = 1(1) + 0(x) + 0(x^2)\). 2. \(T\left(\begin{array}{ll} 0 & 1 \\ 0 & 0 \end{array}\right) = 1 + x^2 = 1(1) + 0(x) + 1(x^2)\). 3. \(T\left(\begin{array}{ll} 0 & 0 \\ 1 & 0 \end{array}\right) = 0 = 0(1) + 0(x) + 0(x^2)\). 4. \(T\left(\begin{array}{ll} 0 & 0 \\ 0 & 1 \end{array}\right) = 2x = 0(1) + 2(x) + 0(x^2)\).
03

Determine the matrix representation [T]岬︶掸

Finally, we form the matrix representation [T]岬︶掸 by placing the coefficients from the linear combinations in Step 2 as columns in the matrix: $$ [T]_\beta^\gamma = \left(\begin{array}{cccc} 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 2 \\ 0 & 1 & 0 & 0 \end{array}\right) $$ Thus, the matrix representation of the linear transformation T relative to bases 尾 and 纬 is: $$ [T]_\beta^\gamma = \left(\begin{array}{llll} 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 2 \\ 0 & 1 & 0 & 0 \end{array}\right) $$

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Most popular questions from this chapter

Find a \(3 \times 3\) orthogonal matrix \(P\) whose first two rows are multiples of (a) (1,2,3) and (0,-2,3) (b) \(\quad(1,3,1)\) and (1,0,-1)

Refer to the following matrices: $$A=\left[\begin{array}{rr} 1 & 2 \\ 3 & -4 \end{array}\right], \quad B=\left[\begin{array}{rr} 5 & 0 \\ -6 & 7 \end{array}\right], \quad C=\left[\begin{array}{rrr} 1 & -3 & 4 \\ 2 & 6 & -5 \end{array}\right], \quad D=\left[\begin{array}{rrr} 3 & 7 & -1 \\ 4 & -8 & 9 \end{array}\right]$$ Find \(\quad(a) A^{T}\) (b) \(B^{T}\) \((\mathrm{c})(A B)^{T}\) (d) \(A^{T} B^{T}\). [Note that \(A^{T} B^{T} \neq(A B)^{T}\).]

Let \(A=\left[\begin{array}{ll}5 & 2 \\ 0 & k\end{array}\right] .\) Find all numbers \(k\) for which \(A\) is a root of the polynomial: (a) \(f(x)=x^{2}-7 x+10\) (b) \(g(x)=x^{2}-25\) (c) \(h(x)=x^{2}-4\)

Let \(e_{i}=[0, \ldots, 0,1,0, \ldots, 0],\) where 1 is the ith entry. Show (a) \(e_{P} A=A_{i},\) ith row of \(A\). (b) \(\quad B e^{T}=B^{j}, j\) th column of \(B\). (c) If \(e_{i} A=e_{i} B,\) for each \(i,\) then \(A=B\). (d) \(\operatorname{If} A e_{j}^{T}=B e_{j}^{T},\) for each \(j,\) then \(A=B\).

Let \(\beta\) and \(\gamma\) be the standard ordered bases for \(\mathrm{R}^{n}\) and \(\mathrm{R}^{m}\), respectively. For each linear transformation \(\mathrm{T}: \mathrm{R}^{n} \rightarrow \mathrm{R}^{m}\), compute \([\mathrm{T}]_{\beta}^{\gamma}\). (a) \(\mathrm{T}: \mathrm{R}^{2} \rightarrow \mathrm{R}^{3}\) defined by $\mathrm{T}\left(a_{1}, a_{2}\right)=\left(2 a_{1}-a_{2}, 3 a_{1}+4 a_{2}, a_{1}\right)$. (b) \(\mathrm{T}: \mathrm{R}^{3} \rightarrow \mathrm{R}^{2}\) defined by $\mathrm{T}\left(a_{1}, a_{2}, a_{3}\right)=\left(2 a_{1}+3 a_{2}-a_{3}, a_{1}+a_{3}\right)$. (c) \(\mathrm{T}: \mathrm{R}^{3} \rightarrow R\) defined by \(\mathrm{T}\left(a_{1}, a_{2}, a_{3}\right)=2 a_{1}+a_{2}-3 a_{3}\). (d) \(\mathrm{T}: \mathrm{R}^{3} \rightarrow \mathrm{R}^{3}\) defined by $$ \mathrm{T}\left(a_{1}, a_{2}, a_{3}\right)=\left(2 a_{2}+a_{3},-a_{1}+4 a_{2}+5 a_{3}, a_{1}+a_{3}\right) . $$ (e) \(\mathrm{T}: \mathrm{R}^{n} \rightarrow \mathrm{R}^{n}\) defined by $\mathrm{T}\left(a_{1}, a_{2}, \ldots, a_{n}\right)=\left(a_{1}, a_{1}, \ldots, a_{1}\right)$. (f) \(\mathrm{T}: \mathrm{R}^{n} \rightarrow \mathrm{R}^{n}\) defined by $\mathrm{T}\left(a_{1}, a_{2}, \ldots, a_{n}\right)=\left(a_{n}, a_{n-1}, \ldots, a_{1}\right)$. (g) \(\mathrm{T}: \mathrm{R}^{n} \rightarrow R\) defined by \(\mathrm{T}\left(a_{1}, a_{2}, \ldots, a_{n}\right)=a_{1}+a_{n}\).
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