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.

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.

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter4: Vector Spaces

Section4.6: Rank Of A Matrix And Systems Of Linear Equations

Problem 67E: Let A be an mn matrix where mn whose rank is r. a What is the largest value r can be? b How many...

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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 mill 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 eristence of a basis for a given finite dimension al subspace. For example,
if V is a subspace of R" of dimension d, then there exists vectors {v;}1 in V such that they form a
basis for V.

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