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Are the following statements true or false? 1. The Gram-Schmidt process produces from a linearly independent set {X₁,...,xp} an orthogonal set {V₁,..., Vp} with the property that for each k, the vectors V₁,..., Vk span the same subspace as that spanned by X₁,..., Xk. ? ? ? ? 2. For all vectors u, v ER", we have u v=-v.u. 3. If vectors V₁,..., Vp span a subspace W and if x is orthogonal to each v; for j = 1, ... , p, then x is in W-. 4. 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}.

Are the following statements true or false? 1. The Gram-Schmidt process produces from a linearly independent set {X₁,...,xp} an orthogonal set {V₁,..., Vp} with the property that for each k, the vectors V₁,..., Vk span the same subspace as that spanned by X₁,..., Xk. ? ? ? ? 2. For all vectors u, v ER", we have u v=-v.u. 3. If vectors V₁,..., Vp span a subspace W and if x is orthogonal to each v; for j = 1, ... , p, then x is in W-. 4. 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}.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter5: Orthogonality
Section5.4: Orthogonal Diagonalization Of Symmetric Matrices
Problem 26EQ
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? 1. The Gram-Schmidt process produces from a linearly independent set {₁,...,xp} an orthogonal set {₁,..., Vp} with the property that for each k, the vectors V₁,..., Vk span the same subspace as that spanned by X₁,...,xk. ? ? ? Are the following statements true or false? ? ✓2. For all vectors u, v ER", we have u V = -V. u. 3. If vectors V₁,..., Vp span a subspace W and if x is orthogonal to each v; for j = 1, ... , p, then x is in W-. 4. 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}. 5. If an n x p matrix U has orthonormal columns, then UUTx = x for all x in R".
Transcribed Image Text:? 1. The Gram-Schmidt process produces from a linearly independent set {₁,...,xp} an orthogonal set {₁,..., Vp} with the property that for each k, the vectors V₁,..., Vk span the same subspace as that spanned by X₁,...,xk. ? ? ? Are the following statements true or false? ? ✓2. For all vectors u, v ER", we have u V = -V. u. 3. If vectors V₁,..., Vp span a subspace W and if x is orthogonal to each v; for j = 1, ... , p, then x is in W-. 4. 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}. 5. If an n x p matrix U has orthonormal columns, then UUTx = x for all x in R".
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