Answered: 2. Let (V,+,) be a vector space over… | bartleby

Linear Algebra: A Modern Introduction

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter6: Vector Spaces

Section6.4: Linear Transformations

Problem 24EQ

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2. Let (V,+,) be a vector space over scalar field R. Suppose the set of vectors {v1, v2, v3, v4} is linearly
independent and consider the following set,
S={v1 2v2, V2, V3 V4 - V2, V4}.
Determine if S is linearly independent. If not, remove as few vectors as you can so the resulting set is
linearly independent and justify your choices.

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Let u, v, and w be any three vectors from a vector space V. Determine whether the set of vectors {vu,wv,uw} is linearly independent or linearly dependent.
Let v1, v2, and v3 be three linearly independent vectors in a vector space V. Is the set {v12v2,2v23v3,3v3v1} linearly dependent or linearly independent? Explain.
Find an orthonormal basis for the subspace of Euclidean 3 space below. W={(x1,x2,x3):x1+x2+x3=0}
Determine whether the set R2 with the operations (x1,y1)+(x2,y2)=(x1x2,y1y2) and c(x1,y1)=(cx1,cy1) is a vector space. If it is, verify each vector space axiom; if it is not, state all vector space axioms that fail.
Let S={v1,v2,v3} be a set of linearly independent vectors in R3. Find a linear transformation T from R3 into R3 such that the set {T(v1),T(v2),T(v3)} is linearly dependent.
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.
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 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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