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Elementary Linear Algebra (MindTap Course List)

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter4: Vector Spaces

Section4.1: Vector In R^n

Problem 28E

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Question

Suppose V1, V2, V3 is an orthogonal set of vectors in R5. Let w be a vector in Span(V1, V2, V3) such that
V₁ • V₁ = 14, v2 • V2 = 6, V3 • V3 = 1,
=
then w=
70, w.v2
V₁+
= −30, w · v3
= 1,
|02+ |03.

Let
27
13
-0-9
=
-4
-4
and let W the subspace of R4 spanned by u and v. Find a basis of W₁, the orthogonal complement of W in R4.

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Take this test to review the material in Chapters 4and Chapters 5. After you are finished, check your work against the answers in the back of the book. Write w=(7,2,4) as a linear combination of the vectors v1, v2 and v3 if possible. v1=(2,1,0), v2=(1,1,0), v3=(0,0,6)
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.
Use a software program or a graphing utility to write v as a linear combination of u1, u2, u3, u4, u5 and u6. Then verify your solution. v=(10,30,13,14,7,27) u1=(1,2,3,4,1,2) u2=(1,2,1,1,2,1) u3=(0,2,1,2,1,1) u4=(1,0,3,4,1,2) u5=(1,2,1,1,2,3) u6=(3,2,1,2,3,0)
Find a basis for R2 that includes the vector (2,2).
Consider the vector v=(1,3,0,4). Find u such that a u has the same direction as v and one-half of its length. b u has the direction opposite that of v and twice its length.
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 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.
Suppose V1, V2, V3 is an orthogonal set of vectors in R5. Let w be a vector in Span(V₁, V2, V3) such that V₁ V₁ = 42, V₂ · V₂ = 542, V3 V3 = 36, -126, w - V₂ - 2710, w - V3 w. • V₁ then w= v₁+ = = = -36, V2+ V3.
Find two vectors v1 and v2 whose sum is (-5, -5, 5), where v1 is parallel to a =(0, –1, – 2) while v2 is perpendicular to a =(0, –1, –2). v1 = and v2 = Use "" signes to enteer a vector.
Find vectors u and v whose initial and terminal points are given. Determine whether u and v are equivalent. u: (1,1), (6,11) v: (-2,5), (3,15) u= 10i-5j, v = 10i+5j which are equivalent u= i+10j, v = 10Oi + 5j which are not equivalent u= 5i+10j, v = 5i+10j which are equivalent u= 5i-10j, v= - 5i-10j which are equivalent Ou= - 5i+10j, v = - 5i+10j which are equivalent
Suppose that v₁ = (4, 4, 0, 7), v₂ = (2, -5, 5, 4) and v3 = (-7, 0, 5, 2). Is the following vector in the span{V₁, V2, V3}? (4, -3,-15, -22) The vector is in the span. The vector is not in the span.
17) Let a = [3, -4] and b = [−6, k]. a) Determine the value(s) of k such that a + b = a + 6. What is the relationship between the two vectors for this result to be true? b) Determine the value(s) of k such that a + b < a + b. What is the relationship between the two vectors for this result to be true?

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