2. Let V be a vector space, and let W₁,..., Wk be finite-dimensional subspaces of V.LetW = W₁ + ... + Wk = {w₁ + ... + Wk | W₁ € W₁, ..., Wk € Wk}.(a) Show that dim(W) ≤ dim(W₁) + + dim(Wk).(b) Show that the following conditions are equivalent:i. W₁, ..., Wk are linearly independent.ii. If ß, is a basis for Wi, i = 1,..., k, then BinB = 0 for all i ‡ j, andØB₁ UU Bk is a basis for W.iii. dim(W) = dim(W₁) + + dim(Wk).iv. Given any vector w W, there exist unique w₁ € W₁,..., wk & Wk suchthat w=w₁ + ··· + Wk.v. W; nΣW; = {0} for each j € {1,…,k}.i‡j(If any one of these conditions holds, one often writesW = W₁0 ... WkⒸand call W the direct sum of W₁,..., Wk.)

Answered: Image /qna-images/answer/16aba76b-de7e-4a6f-8b03-4227c282ac01.jpg

Let V be a vector space, and let W₁,. Let W = .... W₁ + + Wk = {w₁ + + wkw₁ € W₁, ..., wk € Wk}. (a) Show that dim(W) ≤ dim(W₁) + + dim(Wk). (b) Show that the following conditions are equivalent: ... We be finite-dimensional subspaces of V. i. W₁,..., We are linearly independent. ii. If , is a basis for Wi, i = = ... B₁ UU Bk is a basis for W. 1,..., k, then in B₁ = 0 for all i ‡ j, and iii. dim(W) = dim(W₁) + + dim(Wk). iv. Given any vector w E W, there exist unique w₁ € W₁,..., Wk € Wk such that w=w₁ + + Wk. v. W₁nW₂ = {0} for each j = {1, k}. i#j (If any one of these conditions holds, one often writes W = W₁0. and call W the direct sum of W₁, ..., Wk.) Wk

Let V be a vector space, and let W₁,. Let W = .... W₁ + + Wk = {w₁ + + wkw₁ € W₁, ..., wk € Wk}. (a) Show that dim(W) ≤ dim(W₁) + + dim(Wk). (b) Show that the following conditions are equivalent: ... We be finite-dimensional subspaces of V. i. W₁,..., We are linearly independent. ii. If , is a basis for Wi, i = = ... B₁ UU Bk is a basis for W. 1,..., k, then in B₁ = 0 for all i ‡ j, and iii. dim(W) = dim(W₁) + + dim(Wk). iv. Given any vector w E W, there exist unique w₁ € W₁,..., Wk € Wk such that w=w₁ + + Wk. v. W₁nW₂ = {0} for each j = {1, k}. i#j (If any one of these conditions holds, one often writes W = W₁0. and call W the direct sum of W₁, ..., Wk.) Wk

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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Question

2. Let V be a vector space, and let W₁,..., Wk be finite-dimensional subspaces of V.
Let
W = W₁ + ... + Wk = {w₁ + ... + Wk | W₁ € W₁, ..., Wk € Wk}.
(a) Show that dim(W) ≤ dim(W₁) + + dim(Wk).
(b) Show that the following conditions are equivalent:
i. W₁, ..., Wk are linearly independent.
ii. If ß, is a basis for Wi, i = 1,..., k, then BinB = 0 for all i ‡ j, and
Ø
B₁ UU Bk is a basis for W.
iii. dim(W) = dim(W₁) + + dim(Wk).
iv. Given any vector w W, there exist unique w₁ € W₁,..., wk & Wk such
that w=w₁ + ··· + Wk.
v. W; nΣW; = {0} for each j € {1,…,k}.
i‡j
(If any one of these conditions holds, one often writes
W = W₁0 ... Wk
Ⓒ
and call W the direct sum of W₁,..., Wk.)

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