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115_solns_4

115_solns_4 - set T v k 1,T v n is a basis for R T Now...

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115A PROBLEM FROM CLASS, OCT 26 Sec 2.2. 16. Let V and W be vector spaces such that dim V = dim W , and let T : V W be linear. Show there exist ordered bases β for V and γ for W such that [ T ] γ β is diagonal. Let dim V = dim W = n . If T is onto, we can simply take any basis β = { v 1 ,...,v n } of V and γ = { T ( v 1 ) ,...,T ( v n ) } is then a basis of W . Then [ T ( v j )] γ = e j , that is, [ T ] γ β is the identity matrix, which is of course diagonal. Now, if T is not onto, we need to be more careful. We proceed as follows. Recall (cf. Theorem 2.5) that, since dim V = dim W , we have that T is onto if and only if T is one-to-one. So, in this case we know that k := nullity( T ) > 0. Let { v 1 ,...,v k } be a basis of N ( T ). Extend this to an ordered basis β := { v 1 ,...,v k ,v k +1 ,...,v n } for V (cf. Corollary 2, pg. 47-48). As in the proof of the Dimension Theorem, the
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Unformatted text preview: set { T ( v k +1 ) ,...,T ( v n ) } is a basis for R ( T ). Now extend this to an ordered basis γ := { w 1 ,...,w k ,T ( v k +1 ) ,...,T ( v n ) } for W . Then [ T ( v j )] γ = ± if 1 ≤ j ≤ k e j if k + 1 ≤ j ≤ n and so [ T ] γ β = ··· . . . . . . . . . 1 . . . ··· 1 is diagonal, as desired. References Friedberg, et. al. Linear Algebra. Department of Mathematics, University of California, Los Angeles, 90095 E-mail address : [email protected] 1...
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