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Problem 8. PREVIEW ONLY -- ANSWERS NOT RECORDED Are the following statements true or false? 1. If W is a subspace of R" and if v is in both W and W¹, then v must be the zero vector. ? ? 2. If vectors V₁,..., Vp span a subspace W and if x is orthogonal to each V, for j = 1,..., p. then x is in W¹.

Problem 8. PREVIEW ONLY -- ANSWERS NOT RECORDED Are the following statements true or false? 1. If W is a subspace of R" and if v is in both W and W¹, then v must be the zero vector. ? ? 2. If vectors V₁,..., Vp span a subspace W and if x is orthogonal to each V, for j = 1,..., p. then x is in W¹.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter2: Systems Of Linear Equations
Section2.3: Spanning Sets And Linear Independence
Problem 21EQ
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Problem 8. PREVIEW ONLY -- ANSWERS NOT RECORDED Are the following statements true or false? ? ? ? ? 1. If W is a subspace of R" and if v is in both W and W¹, then v must be the zero vector. 2. If vectors V₁,..., Vp span a subspace W and if x is orthogonal to each v, for j = 1,...,p, then x is in W¹ 3. For all vectors u, v E R", we have uv-v.u 4. 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. 5. The best approximation to y by elements of a subspace W is given by the vector y projw (y).
Transcribed Image Text:Problem 8. PREVIEW ONLY -- ANSWERS NOT RECORDED Are the following statements true or false? ? ? ? ? 1. If W is a subspace of R" and if v is in both W and W¹, then v must be the zero vector. 2. If vectors V₁,..., Vp span a subspace W and if x is orthogonal to each v, for j = 1,...,p, then x is in W¹ 3. For all vectors u, v E R", we have uv-v.u 4. 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. 5. The best approximation to y by elements of a subspace W is given by the vector y projw (y).
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