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? Are the following statements true or false? 1. For all vectors u, v E R", we have u. v = -v. u.

? Are the following statements true or false? 1. For all vectors u, v E R", we have u. v = -v. u.

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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? ? ? ? ? Are the following statements true or false? 1. For all vectors u, v € R", we have u v = -v. u. 2. If W = Span{X1, X2, X3} with {X1, X2, X3} linearly independent, and if {V₁, V2, V3} is an orthogonal set in W consisting of non-zero vectors, then {V₁, V2, V3} is an orthogonal basis for W. 3. If {V1, V2, V3} is an orthogonal basis for W, then multiplying V3 by a non-zero scalar c gives a new orthogonal basis {V₁, V₂, CV3}. 4. The best approximation to y by elements of a subspace W is given by the vector y – projw (y). 5. If y = Z₁ + Z₂, where Z₁ is in a subspace W and Z2 is in W, then Z₁ must be the orthogonal projection of y onto W.
Transcribed Image Text:? ? ? ? ? Are the following statements true or false? 1. For all vectors u, v € R", we have u v = -v. u. 2. If W = Span{X1, X2, X3} with {X1, X2, X3} linearly independent, and if {V₁, V2, V3} is an orthogonal set in W consisting of non-zero vectors, then {V₁, V2, V3} is an orthogonal basis for W. 3. If {V1, V2, V3} is an orthogonal basis for W, then multiplying V3 by a non-zero scalar c gives a new orthogonal basis {V₁, V₂, CV3}. 4. The best approximation to y by elements of a subspace W is given by the vector y – projw (y). 5. If y = Z₁ + Z₂, where Z₁ is in a subspace W and Z2 is in W, then Z₁ must be the orthogonal projection of y onto W.
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