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Suppose {V1, V2, V3} are linearly independent vectors in a vector space V. Find the number(s) k such that the vectors w₁ = V₁ + kV2, W2 = V2 + kv3 and w3 = V3 + kv₁ are linearly dependent. Justify your answers.

Suppose {V1, V2, V3} are linearly independent vectors in a vector space V. Find the number(s) k such that the vectors w₁ = V₁ + kV2, W2 = V2 + kv3 and w3 = V3 + kv₁ are linearly dependent. Justify your answers.

Advanced Engineering Mathematics
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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Suppose {V1, V2, V3} are linearly independent vectors in a vector space V. Find the number(s) k such that the vectors w₁ = V₁ + kv₂, W2 = V2 + kv3 and w3 = V3+ kv₁ are linearly dependent. Justify your answers. Hint: Assume C₁W₁+C₂W2+C3W3 0 for some scales C1, C2, C3. Express it in terms of lin- ear combinations of V₁, V2, V3. Then, use the linearly independence of {V₁, V2, V3} to get a linear system with k as a parameter and C₁, C2, C3 as unknown variables. {w₁, W2, W3} are linearly dependent if the system only have zero solutions.
Transcribed Image Text:Suppose {V1, V2, V3} are linearly independent vectors in a vector space V. Find the number(s) k such that the vectors w₁ = V₁ + kv₂, W2 = V2 + kv3 and w3 = V3+ kv₁ are linearly dependent. Justify your answers. Hint: Assume C₁W₁+C₂W2+C3W3 0 for some scales C1, C2, C3. Express it in terms of lin- ear combinations of V₁, V2, V3. Then, use the linearly independence of {V₁, V2, V3} to get a linear system with k as a parameter and C₁, C2, C3 as unknown variables. {w₁, W2, W3} are linearly dependent if the system only have zero solutions.
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