Suppose that V is a vector space. For any two subspaces U, W CV, define their sum as U+W = {v € V: 3uU₁3w W₁v=u+w}. (ali Show that if U and W are subspaces of V, then U+W is also a subspace of V (b) (c) Show that U and W are subspaces of U+W. how that U+W is the smallest subspace containing both U and W; that is, if Y is another subspace of V such that both UCY and WCY, then U+WCY.
Suppose that V is a vector space. For any two subspaces U, W CV, define their sum as U+W = {v € V: 3uU₁3w W₁v=u+w}. (ali Show that if U and W are subspaces of V, then U+W is also a subspace of V (b) (c) Show that U and W are subspaces of U+W. how that U+W is the smallest subspace containing both U and W; that is, if Y is another subspace of V such that both UCY and WCY, then U+WCY.
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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Transcribed Image Text:2. Suppose that V is a vector space. For any two subspaces U, W CV, define their sum as
U + W = {v € V: 3uU₁3w W₁v=u+w}.
(ali Show that if U and W are subspaces of V, then U+W is also a subspace of
V
(b)
(c)
Show that U and W are subspaces of U+W.
how that U+W is the smallest subspace containing both U and W; that is,
if Y is a other subspace of V such that both UC Y and WCY, then U+WCY.
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