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**Unformatted text preview: **MAT 442: Test 1 Name: Instructor: J. Jones 1 2 3 4 5 6 7 Total 10 pts. 15 pts. 25 pts. 10 pts. 20 pts. 30 pts. 20 pts. 130 pts. Write your answers on separate paper. Make sure your name is on every sheet. If a problem asks you to prove a result already covered in the course (theorem, example, or exercise), you need to prove it here. You may use a calculator for this test, although it is unlikely to help. Throughout, F is a field. 1. (10 points) Suppose V is a vector space over F . Prove that for all a F , a ~ 0 = ~ where ~ 0 is the zero vector of V . (This is a theorem from the course.) 2. (a) (5 points) Define the term basis . (b) (10 points) Suppose V is a vector space, S a finite spanning set of V , and L a linearly independent subset of S . Prove that V has a basis B with L B S . (This is a theorem from the course. Do not deduce this using another theorem which asserts the existence of a basis.) 3. Suppose V , W , and X are vector spaces, T : V W and U : W X linear...

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