Use the steps below to prove the following relations among the four fundamental subspaces determined by an m x n matrix A. a. Show that Row A is contained in (Nul A)-. (Show that if x is in Row A, then x is orthogonal to every u in Nul A.) b. Suppose rank A = r. Find dim Nul A and dim (Nul A)→, and then deduce from part (a) that Row A = (Nul A)-. [Hint: Study the exercises for Section 6.3.] c. Explain why Col A = (Nul AT )-.

Use the steps below to prove the following relations among the four fundamental subspaces determined by an m x n matrix A. a. Show that Row A is contained in (Nul A)-. (Show that if x is in Row A, then x is orthogonal to every u in Nul A.) b. Suppose rank A = r. Find dim Nul A and dim (Nul A)→, and then deduce from part (a) that Row A = (Nul A)-. [Hint: Study the exercises for Section 6.3.] c. Explain why Col A = (Nul AT )-.

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter4: Vector Spaces

Section4.CR: Review Exercises

Problem 73CR

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