1. Consider the following four vectors in R5. V₁ = 0 2 V3 2 0 3 V4 Let U span {1, U2, U3, U4}. For the purposes of this exercise (and only this one), if you need to row-reduce a matrix, you can have a computer do it and just write down the RREF, as long as you mention the resource(s) you used. (a) Let 6 [4 5 8 0 -7]. Determine whether EU. Explain your reasoning. (b) Determine whether e₁ EU. Explain your reasoning. (c) Find a 5 x 3 matrix A such that im(A) = U, and explain how you know your choice works. Hint. For this and the next part, you may find the result you proved in Written Assignment 1 useful. (d) Find a 5 x 5 matrix B, which has no columns of all zeroes, such that im(B) = U, and explain how you know your choice works.
1. Consider the following four vectors in R5. V₁ = 0 2 V3 2 0 3 V4 Let U span {1, U2, U3, U4}. For the purposes of this exercise (and only this one), if you need to row-reduce a matrix, you can have a computer do it and just write down the RREF, as long as you mention the resource(s) you used. (a) Let 6 [4 5 8 0 -7]. Determine whether EU. Explain your reasoning. (b) Determine whether e₁ EU. Explain your reasoning. (c) Find a 5 x 3 matrix A such that im(A) = U, and explain how you know your choice works. Hint. For this and the next part, you may find the result you proved in Written Assignment 1 useful. (d) Find a 5 x 5 matrix B, which has no columns of all zeroes, such that im(B) = U, and explain how you know your choice works.
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter1: Vectors
Section1.2: Length And Angle: The Dot Product
Problem 17EQ
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