6a. Suppose that B is a basis of V. Decompose it as a disjoint union B = B₁ I IIB₂ of non-empty sets B₁. Show that B, is a basis of W; := Span(B₂) and V=W₁ 0... Ws is a direct sum of nonzero vector subspaces W₁ of V. 6b. Conversely, suppose that V = W₁ W, is a direct sum of nonzero vector subspaces W₁ of V. Let Bį be a basis of W;. Show that B = B₁ III B₂ is a basis of V and a disjoint union of non-empty sets B₁.

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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6a. Suppose that B is a basis of V. Decompose it as a disjoint union
B = B₁ I
IIB₂
of non-empty sets B₁. Show that B, is a basis of W; := Span(B₂) and
V=W₁ 0... Ws
is a direct sum of nonzero vector subspaces W₁ of V.
6b. Conversely, suppose that V = W₁
W, is a direct sum of
nonzero vector subspaces W₁ of V. Let Bį be a basis of W;. Show that
B = B₁ III B₂
is a basis of V and a disjoint union of non-empty sets B₁.
Transcribed Image Text:6a. Suppose that B is a basis of V. Decompose it as a disjoint union B = B₁ I IIB₂ of non-empty sets B₁. Show that B, is a basis of W; := Span(B₂) and V=W₁ 0... Ws is a direct sum of nonzero vector subspaces W₁ of V. 6b. Conversely, suppose that V = W₁ W, is a direct sum of nonzero vector subspaces W₁ of V. Let Bį be a basis of W;. Show that B = B₁ III B₂ is a basis of V and a disjoint union of non-empty sets B₁.
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