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Su11b - MAS 4105 Test 2(16 pts 1 In the following let V W...

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MAS 4105 Test 2 (16 pts) 1. In the following let V , W , and Z be vector spaces of finite dimension with bases α , β , and γ respectively. Let T and U be a linear transformations such that T : V W and U : W Z . a. If the vector space R m is isomorphic to the vector space P n ( R ), what is the relationship between n and m ? b. If U and T are invertible, what is another way of representing ( UT ) - 1 ? c. If A M n × m ( R ) , then L A maps the vector space into the vector space . d. If [ UT ] a b = [ U ] c d [ T ] e f , then which letters, { a, b, c, d, e, f } , correspond to either α or γ ? e. If S : R 3 R 3 with S a b c = a - b b - c c - a , is S an isomorphism? Explain. f. If β = { x 2 + 1 , x, 1 } what is φ β ( x 2 + x + 1)?
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g. If [ T 1 ] β α = a 1 b 1 c 1 d 1 , [ T 2 ] β α = a 2 b 2 c 2 d 2 , and k is a scalar, then what is the matrix [ k T 1 + T 2 ] β α ? h. If S 1 : M 2 × 2 ( R ) P 2 ( R ) and S 2 : P 2 ( R ) M 2 × 2 ( R ) with S 1 a b c d = (a + b)x 2 + (c + d)x + (d - a) and S 2 ( α x 2 + β x + γ ) = α + γ β - γ α + β γ , what is the image of the vector 1 2 3 4 under the transformation S 2 S 1 ? (6 pts) 2. Let V , W , and Z be vector spaces of finite dimension and let T and U be a linear transformations such that T : V W and U : W Z . Prove that if UT is one to one, then T is one to one.
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(8 pts) 3. Let A i M j × j ( R ) such that each A i is invertible. Use induction to prove that ( A 1 A 2 · · · A n ) is invertible and that ( A 1 A 2 · · · A n ) - 1 = A - 1 n · · · A - 1 2 A - 1 1 .
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