1. Suppose that A is an m x n matrix and B is annxk matrix. (a) Give the definition of the matrix-vector product Ax. What must the size of x be for this to make sense? What is the size of Ax? (b) Give the definition of the matrix-matrix product AB. Use your answer to identify the size of the resulting matrix. (c) Give the definition of the column space of A. This is a subspace of RN for some integer N. What is N and why? (d) Give the definition of the nullspace of A. This is a subspace of RN for some integer N. What is N and why? (e) Suppose that m = n and that the column space and nullspadsof A are equal to each other. Show that A2 = 0. Hint: use part (b). (Note that the "0" here represents the n xn matrix with every entry equal to zero.)
1. Suppose that A is an m x n matrix and B is annxk matrix. (a) Give the definition of the matrix-vector product Ax. What must the size of x be for this to make sense? What is the size of Ax? (b) Give the definition of the matrix-matrix product AB. Use your answer to identify the size of the resulting matrix. (c) Give the definition of the column space of A. This is a subspace of RN for some integer N. What is N and why? (d) Give the definition of the nullspace of A. This is a subspace of RN for some integer N. What is N and why? (e) Suppose that m = n and that the column space and nullspadsof A are equal to each other. Show that A2 = 0. Hint: use part (b). (Note that the "0" here represents the n xn matrix with every entry equal to zero.)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:1. Suppose that A is an m x n matrix and B is an n x k matrix.
(a) Give the definition of the matrix-vector product Ax. What must the size of x be
for this to make sense? What is the size of Ax?
(b) Give the definition of the matrix-matrix product AB. Use your answer to identify
the size of the resulting matrix.
(c) Give the definition of the column space of A. This is a subspace of R for some
integer N. What is N and why?
(d) Give the definition of the nullspace of A. This is a subspace of RN for some integer
N. What is N and why?
(e) Suppose that m = n and that the column space and nullspadsof A are equal to each
other. Show that A2 = 0. Hint: use part (b). (Note that the "0" here represents
the n x n matrix with every entry equal to zero.)
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