Find a basis for the column space and the rank of the matrix.7 14 -6 -324 -3 -624 -2 -2-2 -41 -2(a) a basis for the column space0000

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Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter7: Eigenvalues And Eigenvectors

Section7.CM: Cumulative Review

Problem 26CM: Find an orthogonal matrix P such that PTAP diagonalizes the symmetric matrix A=[1331].

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a Find a symmetric matrix B such that B2=A for A=[2112] b Generalize the result of part a by proving that if A is an nn symmetric matrix with positive eigenvalues, then there exists a symmetric matrix B such that B2=A.
Consider an mn matrix A and an np matrix B. Show that the row vectors of AB are in the row space of B and the column vectors of AB are in the column space of A.
Find an orthogonal matrix P such that PTAP diagonalizes the symmetric matrix A=[1331].
Use the Gram-Schmidt orthonormalization process to find an orthogonal matrix P such that PTAP diagonalizes the symmetric matrix A=[022202220].
35. A permutation matrix is a matrix that can be obtained from an identity matrix by interchanging the rows one or more times (that is, by permuting the rows). For the permutation matrices are and the five matrices. (Sec. , Sec. , Sec. ) Given that is a group of order with respect to matrix multiplication, write out a multiplication table for . Sec. 22. Find the center for each of the following groups . c. in Exercise 35 of section 3.1. 32. Find the centralizer for each element in each of the following groups. c. in Exercise 35 of section 3.1 Sec. 5. The elements of the multiplicative group of permutation matrices are given in Exercise of section. Find the order of each element of the group. Sec. 6. Let be the group of permutations matrices as given in Exercise of Section .
A square matrix is called upper triangular if all of the entries below the main diagonal are zero. Thus, the form of an upper triangular matrix is where the entries marked * are arbitrary. A more formal definition of such a matrix . 29. Prove that the product of two upper triangular matrices is upper triangular.

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Find a basis for the column space and the rank of the matrix. 7 14 -6 -3 2 4 -3 -6 2 4 -2 -2 -2 -4 1-2 (a) a basis for the column space 0000