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2. Let W be a finite-dimensional subspace of an inner product space V. Recall we proved in class that given any v EV, there exists a unique we W such that v − w € W¹, and we call this unique w the orthogonal projection of v on W. Now consider the function T: VV which sends each v € V to its orthogonal projection on W. Prove the following statements using the definition alone, that is, do NOT use any formula for computing orthogonal projection. (a) T is a linear transformation. (b) If v E W then T(v) = v. If v € W then T(v) = 0. (c) R(T) W and N(T) = W. Hence dim(W) + dim(W) dim(V) by the Rank-Nullity Theorem. (Remark: another way to prove dim(W)+dim(W+) = dim(V) is to use the result V W W from Textbook Sec. 6.2 Exer- cise 13(d).) The linear transformation T is called the orthogonal projection of V onto W. = =

2. Let W be a finite-dimensional subspace of an inner product space V. Recall we proved in class that given any v EV, there exists a unique we W such that v − w € W¹, and we call this unique w the orthogonal projection of v on W. Now consider the function T: VV which sends each v € V to its orthogonal projection on W. Prove the following statements using the definition alone, that is, do NOT use any formula for computing orthogonal projection. (a) T is a linear transformation. (b) If v E W then T(v) = v. If v € W then T(v) = 0. (c) R(T) W and N(T) = W. Hence dim(W) + dim(W) dim(V) by the Rank-Nullity Theorem. (Remark: another way to prove dim(W)+dim(W+) = dim(V) is to use the result V W W from Textbook Sec. 6.2 Exer- cise 13(d).) The linear transformation T is called the orthogonal projection of V onto W. = =

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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2. Let W be a finite-dimensional subspace of an inner product space V. Recall we proved in class that given any v € V, there exists a unique w EW such that v — w € W¹, and we call this unique w the orthogonal projection of v on W. Now consider the function T: V → V which sends each v € V to its orthogonal projection on W. Prove the following statements using the definition alone, that is, do NOT use any formula for computing orthogonal projection. (a) T is a linear transformation. (b) If v € W then T(v) = v. If v € W- then T(v) = 0. (c) R(T) W and N(T) = W. Hence dim(W) + dim(W¹) = dim(V) by the Rank-Nullity Theorem. (Remark: another way to prove dim(W)+dim(W+) = dim(V) is to use the result V W W from Textbook Sec. 6.2 Exer- cise 13(d).) The linear transformation T is called the orthogonal projection of V onto W. = =
Transcribed Image Text:2. Let W be a finite-dimensional subspace of an inner product space V. Recall we proved in class that given any v € V, there exists a unique w EW such that v — w € W¹, and we call this unique w the orthogonal projection of v on W. Now consider the function T: V → V which sends each v € V to its orthogonal projection on W. Prove the following statements using the definition alone, that is, do NOT use any formula for computing orthogonal projection. (a) T is a linear transformation. (b) If v € W then T(v) = v. If v € W- then T(v) = 0. (c) R(T) W and N(T) = W. Hence dim(W) + dim(W¹) = dim(V) by the Rank-Nullity Theorem. (Remark: another way to prove dim(W)+dim(W+) = dim(V) is to use the result V W W from Textbook Sec. 6.2 Exer- cise 13(d).) The linear transformation T is called the orthogonal projection of V onto W. = =
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