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115apracticeexam1

# 115apracticeexam1 - Practice Hour Exam#1 Math 115A Section...

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Practice Hour Exam #1 Math 115A Section 1 NAME: SCORES: 1. / 20 2. / 20 3. / 10 4. / 10 5. / 20 6. / 20 Total: / 100

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Problem 1 (20 points - 5 points each) (a) State what it means for a subset S of a vector space V to be linearly independent . (b) Define the dimension of a finite-dimensional vector space. (c) Suppose T : V W is a linear transformation. Define the null space of T . (d) Suppose T : V W is a linear transformation. Define the rank of T .
Problem 2 (20 points - 4 points each) For each of the following statements, determine if they are true or false. If they are true, prove them. If they are false, provide a counterexample . (a) dim( P n ( F )) = n . (b) 3 distinct vectors in R 4 must be linearly independent. (c) 3 distinct vectors in R 2 must be linearly dependent.

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(d) If dim( V ) = n and dim( W ) = m and β and γ are ordered bases for V and W respectively, then [ T ] γ β is an m × n -matrix. (e) If T : V W is a linear transformation, where V and W are finite-dimensional vector spaces, then T is 1-1 if and only if T is onto.
Problem 3 (10 points) Let V = { ( a, b ) | a, b R } . For ( a, b ) , ( a 0 , b 0 ) V and c R , define ( a, b ) ( a 0 , b 0 ) := ( a + a 0 , b + b 0 ) and c ( a, b ) := ( 0 if c = 0 ( ca, b c ) if c 6 = 0 Show that ( V, , ) is not a vector space.

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Problem 4 (10 points) Suppose that W 1 and W 2 are subspaces of a finite-

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115apracticeexam1 - Practice Hour Exam#1 Math 115A Section...

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