/*! 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 12 Find the transpose of each matri... [FREE SOLUTION] | 91影视

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Chapter 2: Problem 12

Find the transpose of each matrix: \(A=\left[\begin{array}{rrr}1 & -2 & 3 \\ 7 & 8 & -9\end{array}\right], \quad B=\left[\begin{array}{lll}1 & 2 & 3 \\ 2 & 4 & 5 \\ 3 & 5 & 6\end{array}\right], \quad C=[1,-3,5,-7], \quad D=\left[\begin{array}{r}2 \\\ -4 \\ 6\end{array}\right]\)

Short Answer

Expert verified
The transposes of the given matrices are: \[A^T=\left[\begin{array}{rr}1 & 7 \\\ -2 & 8 \\\ 3 & -9\end{array}\right],\] \[B^T=\left[\begin{array}{lll}1 & 2 & 3 \\\ 2 & 4 & 5 \\\ 3 & 5 & 6\end{array}\right],\] \[C^T=\left[\begin{array}{r}1 \\\ -3 \\\ 5 \\\ -7\end{array}\right],\] \[D^T=[2, -4, 6]\]

Step by step solution

01

Find the transpose of matrix A

Matrix A is a 2x3 matrix: \[A=\left[\begin{array}{rrr}1 & -2 & 3 \\\ 7 & 8 & -9\end{array}\right]\] To find the transpose of matrix A, we will exchange its rows and columns: \[A^T=\left[\begin{array}{rr}1 & 7 \\\ -2 & 8 \\\ 3 & -9\end{array}\right]\]
02

Find the transpose of matrix B

Matrix B is a 3x3 matrix: \[B=\left[\begin{array}{lll}1 & 2 & 3 \\\ 2 & 4 & 5 \\\ 3 & 5 & 6\end{array}\right]\] To find the transpose of matrix B, we will exchange its rows and columns: \[B^T=\left[\begin{array}{lll}1 & 2 & 3 \\\ 2 & 4 & 5 \\\ 3 & 5 & 6\end{array}\right]\]
03

Find the transpose of matrix C

Matrix C is a row matrix: \[C=[1,-3,5,-7]\] To find the transpose of matrix C, we will exchange its rows and columns: \[C^T=\left[\begin{array}{r}1 \\\ -3 \\\ 5 \\\ -7\end{array}\right]\]
04

Find the transpose of matrix D

Matrix D is a column matrix: \[D=\left[\begin{array}{r}2 \\\ -4 \\\ 6\end{array}\right]\] To find the transpose of matrix D, we will exchange its rows and columns: \[D^T=[2, -4, 6]\] So we have found the transpose of all given matrices: \[A^T=\left[\begin{array}{rr}1 & 7 \\\ -2 & 8 \\\ 3 & -9\end{array}\right],\] \[B^T=\left[\begin{array}{lll}1 & 2 & 3 \\\ 2 & 4 & 5 \\\ 3 & 5 & 6\end{array}\right],\] \[C^T=\left[\begin{array}{r}1 \\\ -3 \\\ 5 \\\ -7\end{array}\right],\] \[D^T=[2, -4, 6]\]

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Most popular questions from this chapter

Let \(\mathrm{V}=\mathrm{P}_{n}(F)\), and let \(c_{0}, c_{1}, \ldots, c_{n}\) be distinct scalars in \(F\). (a) For \(0 \leq i \leq n\), define \(\mathrm{f}_{i} \in \mathrm{V}^{*}\) by \(\mathrm{f}_{i}(p(x))=p\left(c_{i}\right)\). Prove that \(\left\\{\mathrm{f}_{0}, \mathrm{f}_{1}, \ldots, \mathrm{f}_{n}\right\\}\) is a basis for \(\mathrm{V}^{*}\). Hint: Apply any linear combination of this set that equals the zero transformation to \(p(x)=\) \(\left(x-c_{1}\right)\left(x-c_{2}\right) \cdots\left(x-c_{n}\right)\), and deduce that the first coefficient is zero. (b) Use the corollary to Theorem \(2.26\) and (a) to show that there exist unique polynomials \(p_{0}(x), p_{1}(x), \ldots, p_{n}(x)\) such that \(p_{i}\left(c_{j}\right)=\delta_{i j}\) for \(0 \leq i \leq n\). These polynomials are the Lagrange polynomials defined in Section 1.6. (c) For any scalars \(a_{0}, a_{1}, \ldots, a_{n}\) (not necessarily distinct), deduce that there exists a unique polynomial \(q(x)\) of degree at most \(n\) such that \(q\left(c_{i}\right)=a_{i}\) for \(0 \leq i \leq n .\) In fact, $$ q(x)=\sum_{i=0}^{n} a_{i} p_{i}(x) $$ (d) Deduce the Lagrange interpolation formula: $$ p(x)=\sum_{i=0}^{n} p\left(c_{i}\right) p_{i}(x) $$ for any \(p(x) \in \mathrm{V}\). (e) Prove that $$ \int_{a}^{b} p(t) d t=\sum_{i=0}^{n} p\left(c_{i}\right) d_{i} $$ where $$ d_{i}=\int_{a}^{b} p_{i}(t) d t $$ Suppose now that $$ c_{i}=a+\frac{i(b-a)}{n} \text { for } i=0,1, \ldots, n \text {. } $$ For \(n=1\), the preceding result yields the trapezoidal rule for evaluating the definite integral of a polynomial. For \(n=2\), this result yields Simpson's rule for evaluating the definite integral of a polynomial.

Assume the notation in Theorem \(2.13 .\) (a) Suppose that \(z\) is a (column) vector in \(\mathrm{F}^{p}\). Use Theorem 2.13(b) to prove that \(B z\) is a linear combination of the columns of \(B\). In particular, if \(z=\left(a_{1}, a_{2}, \ldots, a_{p}\right)^{t}\), then show that $$ B z=\sum_{j=1}^{p} a_{j} v_{j} . $$ (b) Extend (a) to prove that column \(j\) of \(A B\) is a linear combination of the columns of \(A\) with the coefficients in the linear combination being the entries of column \(j\) of \(B\). (c) For any row vector \(w \in \mathrm{F}^{m}\), prove that \(w A\) is a linear combination of the rows of \(A\) with the coefficients in the linear combination being the coordinates of \(w\). Hint: Use properties of the transpose operation applied to (a). (d) Prove the analogous result to (b) about rows: Row \(i\) of \(A B\) is a linear combination of the rows of \(B\) with the coefficients in the linear combination being the entries of row \(i\) of \(A\).

Prove the following generalization of Theorem 2.23. Let $\mathrm{T}: \mathrm{V} \rightarrow \mathrm{W}$ be a linear transformation from a finite- dimensional vector space \(V\) to a finite-dimensional vector space W. Let \(\beta\) and \(\beta^{\prime}\) be ordered bases for \(\mathrm{V}\), and let \(\gamma\) and \(\gamma^{\prime}\) be ordered bases for \(\mathrm{W}\). Then $[\mathrm{T}]_{\beta}^{\gamma}^{\prime}=P^{-1}[\mathrm{~T}]_{\beta}^{\gamma} Q\(, where \)Q\( is the matrix that changes \)\beta^{\prime}$-coordinates into \(\beta\)-coordinates and \(P\) is the matrix that changes \(\gamma^{\prime}\)-coordinates into \(\gamma\)-coordinates.

Let \(A\) be an \(n \times n\) matrix. (a) Suppose that \(A^{2}=O .\) Prove that \(A\) is not invertible. (b) Suppose that \(A B=O\) for some nonzero \(n \times n\) matrix \(B\). Could \(A\) be invertible? Explain.

For each of the following vector spaces \(V\) and bases \(\beta\), find explicit formulas for vectors of the dual basis \(\beta^{*}\) for \(\mathrm{V}^{*}\), as in Example 4 . (a) \(\mathrm{V}=\mathrm{R}^{3} ; \beta=\\{(1,0,1),(1,2,1),(0,0,1)\\}\) (b) \(\mathrm{V}=\mathrm{P}_{2}(R) ; \beta=\left\\{1, x, x^{2}\right\\}\)
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