5. Let V be a vector space over R, and suppose that v1, V2, V3, V4 are linearly independent vectors in V.a. Prove that the vectorsV1 – V3,2v2 – 3v3 + V4,-v1 + 4v2 + 2v3 + v4,vị + 2v2 + 3v3are linearly dependent.b. Prove that the vectorsV1 + v2 + V3 + V4,V1 – V2,V1 + v2 – 2v3 + v4,3v3 – V4are linearly independent.

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5. Let V be a vector space over R, and suppose that v1, v2, v3, V4 are linearly independent vectors in V. a. Prove that the vectors Vị – V3, 2v2 – 3v3 + V4, -vi + 4v2 + 2v3 + v4, vị + 2v2 + 3v3 are linearly dependent. b. Prove that the vectors vị + v2 + V3 + v4, Vi – V2, vi + v2 – 2v3 + v4, 3v3 – V4 are linearly independent.

5. Let V be a vector space over R, and suppose that v1, v2, v3, V4 are linearly independent vectors in V. a. Prove that the vectors Vị – V3, 2v2 – 3v3 + V4, -vi + 4v2 + 2v3 + v4, vị + 2v2 + 3v3 are linearly dependent. b. Prove that the vectors vị + v2 + V3 + v4, Vi – V2, vi + v2 – 2v3 + v4, 3v3 – V4 are linearly independent.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.

Question

5. Let V be a vector space over R, and suppose that v1, V2, V3, V4 are linearly independent vectors in V.
a. Prove that the vectors
V1 – V3,
2v2 – 3v3 + V4,
-v1 + 4v2 + 2v3 + v4,
vị + 2v2 + 3v3
are linearly dependent.
b. Prove that the vectors
V1 + v2 + V3 + V4,
V1 – V2,
V1 + v2 – 2v3 + v4,
3v3 – V4
are linearly independent.

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