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# hw5 - X i =1-i = 1-1 Now let T R R be a linear...

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HOMEWORK 5 For all of the following exercises, decide if the statement is true or false, and then prove your decision correct. 1. Let V = M 5 × 5 ( C ), W = P 24 ( C ), and T : V W be a linear transformation. Then T is an isomorphism. 2. If V and W are ﬁnite dimensional vector spaces, and T : V W is linear, then the following are equivalent: T is one-to-one. T is surjective. T is an isomorphism. 3. There exist inﬁnite dimensional vector spaces V and W , and a linear transformation T : V W , such that the following statement is false: T is injective iﬀ null( T ) = 0 4. Let F p be the ﬁnite ﬁeld with p elements (where p is some prime number), and let n be some positive integer. Then |L ( F n p , F p ) | = p n . 5. It’s not hard to show that:
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Unformatted text preview: X i =1 -i = 1 -1 Now, let T : R R be a linear transformation with R regarded as a Q-vector space (i.e. T is additive, and homogeneous with respect to rational scalars). Then: X i =1 T ( -i ) = T 1 -1 [NOTE: Lets say that a real number r is transcendental i 6 p ( x ) P ( Q )( p ( r ) = 0). If necessary, you may use, without proof, the fact that is transcendental.] 6. If A and B are ( m n )- and ( n m )-matrices, respectively, then even though AB and BA have dierent dimensions, both are square, and det( AB ) = det( BA )....
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