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Linear Algebra: A Modern Introduction

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter5: Orthogonality

Section5.1: Orthogonality In Rn

Problem 7EQ

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Question

Are the following statements true or false?
True
False
1. Three nonzero vectors that lie in a plane in R³ might form a basis for R³.
2. If the set of vectors U spans a subspace S, then vectors can be removed from U to create a basis for S
False ✓ 3. If S = span{u1, U2, U3}, then dim(S) = 3 .
False 4. If the set of vectors U is linearly independent in a subspace S then vectors can be added to U to create a basis for S
False ✓ 5. If S₁ and S₂ are subspaces of R" of the same dimension, then S₁ = S₂.

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Find a basis for R3 that includes the vector (1,0,2) and (0,1,1).
Complete Example 2 by verifying that {1,x,x2,x3} is an orthonormal basis for P3 with the inner product p,q=a0b0+a1b1+a2b2+a3b3. An Orthonormal basis for P3. In P3, with the inner product p,q=a0b0+a1b1+a2b2+a3b3 The standard basis B={1,x,x2,x3} is orthonormal. The verification of this is left as an exercise See Exercise 17..
Take this test to review the material in Chapters 4 and 5. After you are finished, check your work against the answers in the back of the book. Prove that the set of all singular 33 matrices is not a vector space.
Prove that in a given vector space V, the zero vector is unique.
Consider the vectors u=(6,2,4) and v=(1,2,0) from Example 10. Without using Theorem 5.9, show that among all the scalar multiples cv of the vector v, the projection of u onto v is the closest to u that is, show that d(u,projvu) is a minimum.
In Exercises 7-10, show that the given vectors form an orthogonal basis for2or3. Then use Theorem 5.2 to express was a linear combination of these basis vectors. Give the coordinate vector[w] ofwwith respect to the basis ={v1,v2}of 2or =v1,v2,v3 of3. v1=[111],v2=[110],v3=[112];w=[123]

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