Can someone explain why the smaller et spans O W3 n the Corollary below orollary 1.7.1. If {v}} and {w,} are a basis of V and W respectively, then {v; ® w;} is a basis of VW. oof. Since the vectors of the form v w, ve V, w e W, span V W, the much smaller set {v; w;} also ans V W. But dim(V & W) = dim V dim W is precisely the number of elements in the set {v; ® w;}. nce the set {v; w;} is a basis.
Can someone explain why the smaller et spans O W3 n the Corollary below orollary 1.7.1. If {v}} and {w,} are a basis of V and W respectively, then {v; ® w;} is a basis of VW. oof. Since the vectors of the form v w, ve V, w e W, span V W, the much smaller set {v; w;} also ans V W. But dim(V & W) = dim V dim W is precisely the number of elements in the set {v; ® w;}. nce the set {v; w;} is a basis.
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:Can someone explain why the smaller
set spans
V ® W³
in the Corollary below
Corollary 1.7.1. If {vz} and {w;} are a basis of V and W respectively, then {v; ® w;} is a basis of V W.
Proof. Since the vectors of the form v w, ve V, w e W, span V W, the much smaller set {v; 8 w;} also
spans V & W3. But dim(V & W) = dim V • dim W is precisely the number of elements in the set {v; & w;}.
Hence the set {v; ® w;} is a basis.
%3D
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