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3. Let W be a subspace of R" of dimension k < n. Use the Fundamental Theorem of Linear Algebra to prove that W is equivalent to the null space of some matrix. Note that this implies that W is an intersection of hyperplanes in R" that also intersect the origin. Hint: You may assume the existence of a basis for a given finite dimensional subspace. For example, if V is a subspace of Rm of dimension d, then there exists vectors {v;}-1 in V such that they form a basis for V.

3. Let W be a subspace of R" of dimension k < n. Use the Fundamental Theorem of Linear Algebra to prove that W is equivalent to the null space of some matrix. Note that this implies that W is an intersection of hyperplanes in R" that also intersect the origin. Hint: You may assume the existence of a basis for a given finite dimensional subspace. For example, if V is a subspace of Rm of dimension d, then there exists vectors {v;}-1 in V such that they form a basis for V.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter4: Eigenvalues And Eigenvectors
Section4.2: Determinants
Problem 10AEXP
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2. Let u e R™ and v e Rn be two non-zero vectors, in other words at least one component of the vectors is non-zero. Let A = uv' E R™Xn. (a) Suppose ||u||, = 1 and ||v|l2 = 1. Show that the Frobenius norm of A is equal to 1. (b) Consider the case where m = 3 and n = 2, i.e., -E) --(:) u = U2 v = V2 To help simplify your work in the following subproblems you may assume ui 70 and vi # 0. i. Derive a basis for the range of A using Gaussian elimination. What is the rank of A? ii. Derive a basis for the null space of A using Gaussian elimination. (c) Now consider the general case where m and n are any positive integers. To help simplify your work in the following subproblems you may assume u1 # 0 and vi 7 0. i. Generalize your work from b.i to derive a basis for the range of A. What is the rank of A? ii. Generalize your work from b.ii to derive a basis for the null space of A. 3. Let W be a subspace of R" of dimension k < n. Use the Fundamental Theorem of Linear Algebra to prove that W is equivalent to the null space of some matrix. Note that this implies that W is an intersection of hyperplanes in R" that also intersect the origin. Hint: You may assume the existence of a basis for a given finite dimensional subspace. For example, if V is a subspace of Rm of dimension d, then there exists vectors {v;}-1 in V such that they form a basis for V.
Transcribed Image Text:2. Let u e R™ and v e Rn be two non-zero vectors, in other words at least one component of the vectors is non-zero. Let A = uv' E R™Xn. (a) Suppose ||u||, = 1 and ||v|l2 = 1. Show that the Frobenius norm of A is equal to 1. (b) Consider the case where m = 3 and n = 2, i.e., -E) --(:) u = U2 v = V2 To help simplify your work in the following subproblems you may assume ui 70 and vi # 0. i. Derive a basis for the range of A using Gaussian elimination. What is the rank of A? ii. Derive a basis for the null space of A using Gaussian elimination. (c) Now consider the general case where m and n are any positive integers. To help simplify your work in the following subproblems you may assume u1 # 0 and vi 7 0. i. Generalize your work from b.i to derive a basis for the range of A. What is the rank of A? ii. Generalize your work from b.ii to derive a basis for the null space of A. 3. Let W be a subspace of R" of dimension k < n. Use the Fundamental Theorem of Linear Algebra to prove that W is equivalent to the null space of some matrix. Note that this implies that W is an intersection of hyperplanes in R" that also intersect the origin. Hint: You may assume the existence of a basis for a given finite dimensional subspace. For example, if V is a subspace of Rm of dimension d, then there exists vectors {v;}-1 in V such that they form a basis for V.
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