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Midterm1 - Midterm Solutions J Scholtz Question 1 True vs...

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Midterm Solutions J. Scholtz October 29, 2006 Question 1: True vs. False 1. Question: Let W 1 , W 2 be subspaces of V and β 1 , β 2 be their respective basis. Then is it true that β 1 β 2 is a basis for W 1 W 2 ? Answer: No, consider W 1 = W 2 = R R . Let β 1 = - 1 and β 2 = 1. Then β 1 β 2 = 0 and so it is not a basis for W 1 W 2 = R . 2. Question: Let P ( R ) denote the infinite dimensional vector space of all polynomials with real ceficients. Let g ( x ) P ( R ) be some fixed polynomial. Then the function T : P ( R ) P ( R ), such that T ( f ( x )) = g ( x ) f ( x ) is linear. Answer: Yes, we can check that T ( c × a ( x ) + b ( x )) = g ( x )( c × a ( x ) + b ( x )) = g ( x )( c × a ( x )) + g ( x ) b ( x ) = c × T ( a ( x )) + T ( b ( x )). 3. Question: Let V be a vector space and T, U : V V be two linear operators. Then N ( T ) N ( TU ). Answer: No. Consider V = R 2 and the maps T ( x, y ) = ( x, 0) and U ( x, y ) = ( y, x ). Then N ( T ) = span((0 , 1)), whereas N ( TU ) = span((1 , 0)). 4. Question: Let V be a vector space and T, U : V V be two linear operators. Then R ( TU ) R ( T ). Answer: Yes. R ( TU ) = T ( U ( V ))) = T ( R ( U )) T ( V ) since U ( V ) = R ( U ) V .
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