17. Determine whether each statement is true or false, and provide a justifica- tion or a counterexample. a) Suppose that 71, 72, , Um are linear independent in a vector space V. Let EV such that w, U are linear independent for every k 1,2,...,m. Then, 71, 72, , Um, are also linearly independent. = b) Suppose that 7₁, 72, 73 are linear independent in a vector space V. Then, U1U2, U2U3, 73 - 7₁ are also linearly independent. c) Suppose that 7₁, 72, 73 are linear independent in a vector space V. Then, V₁ + V2, V2 + V3, V3 +7₁ are also linearly independent.
17. Determine whether each statement is true or false, and provide a justifica- tion or a counterexample. a) Suppose that 71, 72, , Um are linear independent in a vector space V. Let EV such that w, U are linear independent for every k 1,2,...,m. Then, 71, 72, , Um, are also linearly independent. = b) Suppose that 7₁, 72, 73 are linear independent in a vector space V. Then, U1U2, U2U3, 73 - 7₁ are also linearly independent. c) Suppose that 7₁, 72, 73 are linear independent in a vector space V. Then, V₁ + V2, V2 + V3, V3 +7₁ are also 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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Transcribed Image Text:17. Determine whether each statement is true or false, and provide a justifica-
tion or a counterexample.
a) Suppose that 71, 72, , Um are linear independent in a vector space V. Let
WE V such that w, k are linear independent for every k 1,2,...,m.
Then, 71, 72, , Um, are also linearly independent.
=
b) Suppose that 7₁, 72, 73 are linear independent in a vector space V. Then,
U1U2, U2₂ V3, 73 - 7₁ are also linearly independent.
c) Suppose that 7₁, 72, 73 are linear independent in a vector space V. Then,
V₁ + V2, V2 + V3, V3+ V₁ are also linearly independent.
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