TRUE OR FALSE 8. The polynomials z-1, (z-1)², and (x-1)³ span R,[r].9. If S and T are finite spanning sets for a vector space V, then S and I have the same numberof elements.10. If {v₁, 2, 3} is linearly independent on V, then {v₁-02, 02-03, 03-0₁} is linearly independent.

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Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter6: Linear Transformations

Section6.1: Introduction To Linear Transformations

Problem 78E: Let S={v1,v2,v3} be a set of linearly independent vectors in R3. Find a linear transformation T from...

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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.
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Let V and W be vector spaces, let T : V --> W be linear, and let { w1 , w2 , ... , wk} be a linearly independent set of k vectors from R(T). Prove that if S = {v1,v2 , .. . ,vk} is chosen so that T(vi) = Wi for i = 1, 2, ... , k, then S is linearly independent.
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7.4 Let F(x,y,z)=(−7xz2,9xyz,−2xy3z)F(x,y,z)=(−7xz2,9xyz,−2xy3z) be a vector field and f(x,y,z)=x3y2zf(x,y,z)=x3y2z.∇f=(∇f=( , , )).∇×F=(∇×F=( , , )).F×∇f=(F×∇f=( , , )).F⋅∇f=F⋅∇f=
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The polynomials 1-1, (1-1)², and (x-1)³ span R3[1]. I