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3. Let W be a subspace of R" of dimension k s 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 s 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.

Advanced Engineering Mathematics
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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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.
Transcribed Image Text: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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