Q03. be a basis of V. Let V be an n-dimensional vector space over F, with n ≥ 1. Let B = {v₁,..., Vn} 1. (a) Let *x € V. Prove that if [x]B="0F, then "x = "0y. 2. (b) Let S be a subspace of V, let C={w 1..w k}be a basis of S, and let T={[v]B:"VES} be a subspace of Fn. Prove that dim T = k.
Q03. be a basis of V. Let V be an n-dimensional vector space over F, with n ≥ 1. Let B = {v₁,..., Vn} 1. (a) Let *x € V. Prove that if [x]B="0F, then "x = "0y. 2. (b) Let S be a subspace of V, let C={w 1..w k}be a basis of S, and let T={[v]B:"VES} be a subspace of Fn. Prove that dim T = k.
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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![Q03.
Let V be an n-dimensional vector space over F, with n 2 1. Let B = {°v1, . ,*Vn}
be a basis of V.
1. (а)
2. (b)
Let *x E V. Prove that if ["x]B = Of», then *x = "0v.
Let S be a subspace of V, let C={w1..,w°k}be a basis of S, and let T={[°v]p:°V€S}
be a subspace of F". Prove that dim T = k.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2aaa4768-25f6-4170-96d1-ff5e8e01e567%2Fcd8947df-9a49-4fbd-8b8f-26402e7d412c%2Fm0dv7zn_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Q03.
Let V be an n-dimensional vector space over F, with n 2 1. Let B = {°v1, . ,*Vn}
be a basis of V.
1. (а)
2. (b)
Let *x E V. Prove that if ["x]B = Of», then *x = "0v.
Let S be a subspace of V, let C={w1..,w°k}be a basis of S, and let T={[°v]p:°V€S}
be a subspace of F". Prove that dim T = k.
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