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5. Let V₁, V2, V3, and v4 be such that {V1, V2, V3} is linearly dependent and {v2, V3, V4} is linearly independent. (a) Prove that v₁ is a linear combination of v2 and v3. You can use the fact that any subset of a linearly independent set is linearly independent. Be careful not to assume something is nonzero. (b) Prove that v4 is not a linear combination of V1, V2, and v3.

5. Let V₁, V2, V3, and v4 be such that {V1, V2, V3} is linearly dependent and {v2, V3, V4} is linearly independent. (a) Prove that v₁ is a linear combination of v2 and v3. You can use the fact that any subset of a linearly independent set is linearly independent. Be careful not to assume something is nonzero. (b) Prove that v4 is not a linear combination of V1, V2, and v3.

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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5. Let **v₁, v₂, v₃,** and **v₄** be such that \(\{ \mathbf{v_1}, \mathbf{v_2}, \mathbf{v_3} \}\) is linearly dependent and \(\{ \mathbf{v_2}, \mathbf{v_3}, \mathbf{v_4} \}\) is linearly independent. (a) Prove that **v₁** is a linear combination of **v₂** and **v₃**. You can use the fact that any subset of a linearly independent set is linearly independent. Be careful not to assume something is nonzero. (b) Prove that **v₄** is *not* a linear combination of **v₁, v₂,** and **v₃**.
Transcribed Image Text:5. Let **v₁, v₂, v₃,** and **v₄** be such that \(\{ \mathbf{v_1}, \mathbf{v_2}, \mathbf{v_3} \}\) is linearly dependent and \(\{ \mathbf{v_2}, \mathbf{v_3}, \mathbf{v_4} \}\) is linearly independent. (a) Prove that **v₁** is a linear combination of **v₂** and **v₃**. You can use the fact that any subset of a linearly independent set is linearly independent. Be careful not to assume something is nonzero. (b) Prove that **v₄** is *not* a linear combination of **v₁, v₂,** and **v₃**.
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