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Let V be a finite-dimensional inner product space. Let WCV be a subspace, and let P: V→ V be the orthogonal projection onto W. The reflection about W is the linear operator T: V→ V defined by T(v) = v + 2(P(v) — v) = 2P(v) — v for all v € V. (a) Show that the reflection about W is an orthogonal/unitary operator. (b) Show that if V is a real inner product space, then the reflection about Wis orientation-preserving orientation-reversing if dim V - dim W is even; if dim V dim W is odd.

Let V be a finite-dimensional inner product space. Let WCV be a subspace, and let P: V→ V be the orthogonal projection onto W. The reflection about W is the linear operator T: V→ V defined by T(v) = v + 2(P(v) — v) = 2P(v) — v for all v € V. (a) Show that the reflection about W is an orthogonal/unitary operator. (b) Show that if V is a real inner product space, then the reflection about Wis orientation-preserving orientation-reversing if dim V - dim W is even; if dim V dim W is odd.

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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2. Let V be a finite-dimensional inner product space. Let WCV be a subspace, and let P: VV be the orthogonal projection onto W. The reflection about W is the linear operator T: V→ V defined by T(v): = v + 2(P(v) — v) = 2P(v) — v - for all v € V. (a) Show that the reflection about W is an orthogonal/unitary operator. (b) Show that if V is a real inner product space, then the reflection about W is orientation-preserving orientation-reversing if dim V dim W is even; if dim V - dim W is odd.
Transcribed Image Text:2. Let V be a finite-dimensional inner product space. Let WCV be a subspace, and let P: VV be the orthogonal projection onto W. The reflection about W is the linear operator T: V→ V defined by T(v): = v + 2(P(v) — v) = 2P(v) — v - for all v € V. (a) Show that the reflection about W is an orthogonal/unitary operator. (b) Show that if V is a real inner product space, then the reflection about W is orientation-preserving orientation-reversing if dim V dim W is even; if dim V - dim W is odd.
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