1) Let V be finite-dimensional. Let W1, · · , Wk C V be subspaces, and sup- ... k pose that V Wi. Let dim(W;) = ni and let B; {vi, · .. , v } be a basis Nį. i=1 for W;. Prove that ß; N B; = Ø for all i + j, and that B = | JB; is a basis for V.
1) Let V be finite-dimensional. Let W1, · · , Wk C V be subspaces, and sup- ... k pose that V Wi. Let dim(W;) = ni and let B; {vi, · .. , v } be a basis Nį. i=1 for W;. Prove that ß; N B; = Ø for all i + j, and that B = | JB; is a basis for V.
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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please prove
Transcribed Image Text:Your answers to the problems in this section should be proofs, unless otherwise
stated. F is a field, V and W are vector spaces over F.
1) Let V be finite-dimensional. Let W1, ·.., Wk C V be subspaces, and sup-
k
pose that =
AW;. Let dim(W;)
= n; and let B; =
{vi, ... , v} be a basis
ni
i=1
k
for W;. Prove that B; n B;
= Ø for all i j, and that B = Uß; is a basis for V.
i=1
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