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Prove the following result: Let V be a finite-dimensional vector space, let V₁, V2, ..., Un be vectors in V, and let {1, 2,..., n} be a basis for V' with the property that þ; (vi) = d¿j for all i, j = 1, 2, ..., n. Prove that {v₁, V₂,..., Un} is a basis for V.. Then find an explicit formula for writing an arbitrary u € V as a linear combination of V₁, V2,..., Un.

Prove the following result: Let V be a finite-dimensional vector space, let V₁, V2, ..., Un be vectors in V, and let {1, 2,..., n} be a basis for V' with the property that þ; (vi) = d¿j for all i, j = 1, 2, ..., n. Prove that {v₁, V₂,..., Un} is a basis for V.. Then find an explicit formula for writing an arbitrary u € V as a linear combination of V₁, V2,..., Un.

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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Prove the following result: Let V be a finite-dimensional vector space, let V₁, V2, ..., Un be vectors in V, and let {1, 2,..., n} be a basis for V' with the property that þ; (vi) = d¿j for all i, j = 1, 2, ..., n. Prove that {v₁, V₂,..., Un} is a basis for V.. Then find an explicit formula for writing an arbitrary u € V as a linear combination of V₁, V2,..., Un.
Transcribed Image Text:Prove the following result: Let V be a finite-dimensional vector space, let V₁, V2, ..., Un be vectors in V, and let {1, 2,..., n} be a basis for V' with the property that þ; (vi) = d¿j for all i, j = 1, 2, ..., n. Prove that {v₁, V₂,..., Un} is a basis for V.. Then find an explicit formula for writing an arbitrary u € V as a linear combination of V₁, V2,..., Un.
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