# Problem 5 20 pts true or false to discourage guessing

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Problem 5 (20 pts). True or False? To discourage guessing, the problem will be graded as follows: 2 pts for each correct answer. 0 pts for a blank. -2 pts for each incorrect answer. If anything else but “True” or “False” is written, more than one answer is written, or the answer is hard to read, you will get -2 point. Note: In all questions below V and W are vector spaces over a field F . They may or may not be finite-dimensional. Thus, read the questions carefully and do not assume anything that is not given. (1) The intersection of any two subsets of V is a subspace of V .
(3) If T : V W is a function with T ( ~v 1 + ~v 2 ) = T ( ~v 1 ) + T ( ~v 2 ) for any ~v 1 ,~v 2 V , then T is linear .
(4) If T : V W is a linear transformation, dim V = m , dim W = n , and β and γ are ordered bases of V and W , respectively, then [ T ] γ β is an m × n matrix.
Answer: (6) The Replacement Theorem is used, among other places, in the proof that the dimension of a finite-dimensional vector space is a well-defined concept.
Answer: (8) There are at least three different linear transformations T : R 2 R 2 such that T 2 = T .
GOOD LUCK! 8