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Let W C R" be a subspace and let W- C R" be its orthogonal complement. Let S {u1,..., uk} be an orthogonal basis for W and S' The goal of these exercises is to give two different arguments for the fact that {V1,..., vi} be an othogonal basis for W-. dim W + dim W- = n.

Let W C R" be a subspace and let W- C R" be its orthogonal complement. Let S {u1,..., uk} be an orthogonal basis for W and S' The goal of these exercises is to give two different arguments for the fact that {V1,..., vi} be an othogonal basis for W-. dim W + dim W- = n.

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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5. Let p = i o projw : R" → W → R" be the function defined by orthogonal projection onto W followed by inclusion, i.e. i(x) = x for all x E W. Show that the null space of p Null(p) := {v € R" : p(v) = 0} is exactly W. Now explain why and how one can deduce (*) from this result and the rank theorem.
Transcribed Image Text:5. Let p = i o projw : R" → W → R" be the function defined by orthogonal projection onto W followed by inclusion, i.e. i(x) = x for all x E W. Show that the null space of p Null(p) := {v € R" : p(v) = 0} is exactly W. Now explain why and how one can deduce (*) from this result and the rank theorem.
Let W C R" be a subspace and let W- C R" be its orthogonal complement. Let S {u1,..., uk} be an orthogonal basis for W and S' The goal of these exercises is to give two different arguments for the fact that {V1,..., Vi} be an othogonal basis for W-. (*) dim W + dim W- = n.
Transcribed Image Text:Let W C R" be a subspace and let W- C R" be its orthogonal complement. Let S {u1,..., uk} be an orthogonal basis for W and S' The goal of these exercises is to give two different arguments for the fact that {V1,..., Vi} be an othogonal basis for W-. (*) dim W + dim W- = n.
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