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Linear Algebra: A Modern Introduction

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter1: Vectors

Section: Chapter Questions

Problem 20RQ

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Question

1
2
as the sum of {ủ, 7, w}.
3
Write i
||
4

1
1
9. We have a subspace W in R“ spanned by the following three vectors i =
-2
Show that {ū, ī, w} is a basis of W. Compute their inner products.
3
and w =
-2

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Find a basis for R3 that includes the vector (1,0,2) and (0,1,1).
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.
Find an orthonormal basis for the subspace of Euclidean 3 space below. W={(x1,x2,x3):x1+x2+x3=0}
Use the inner product u,v=2u1v1+u2v2 in R2 and Gram-Schmidt orthonormalization process to transform {(2,1),(2,10)} into an orthonormal basis.
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.
Let B={(0,2,2),(1,0,2)} be a basis for a subspace of R3, and consider x=(1,4,2), a vector in the subspace. a Write x as a linear combination of the vectors in B.That is, find the coordinates of x relative to B. b Apply the Gram-Schmidt orthonormalization process to transform B into an orthonormal set B. c Write x as a linear combination of the vectors in B.That is, find the coordinates of x relative to B.
Define T:R2R2 by T(v)=projuv Where u is a fixed vector in R2. Show that the eigenvalues of A the standard matrix of T are 0 and 1.
Let V be an two dimensional subspace of R4 spanned by (0,1,0,1) and (0,2,0,0). Write the vector u=(1,1,1,1) in the form u=v+w, where v is in V and w is orthogonal to every vector in V.

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