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MidISample copy - P(b Let R 3 = a(1 x 2 b x-3 2 | a and b...

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Prof. Ming Gu, 861 Evans, tel: 2-3145 Email: [email protected] http://www.math.berkeley.edu/ mgu/MA110F2011 Math110 Sample Midterm I, Fall 2011 This is a closed book exam; but everyone is allowed a standard one-page cheat sheet (on one-side only). You need to justify every one of your answers unless you are asked not to do so. Completely correct answers given without justification will receive little credit. Problems are not necessarily ordered according to difficulties. You need not simplify your answers unless you are specifically asked to do so. Hand in this exam before you leave. Problem Maximum Score Your Score 1 20 2 20 3 20 4 20 5 20 Total 100 Write your personal information below and on top of every page in the test. Your Name: Your GSI: Your SID:
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Name and SID: 2 1. Let P be the set of all polynomials with real coefficients. (a) Let Q 3 be the set of all polynomials of degree exactly 3 with real coefficients. Is Q 3 a subspace of
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Unformatted text preview: P ? (b) Let R 3 = { a (1 + x 2 ) + b ( x-3 2 ) | a and b are real numbers . } . Is R 3 a subspace of P ? Name and SID: 3 2. Let f 1 = 1 1 1 1 , f 2 = 1 1 1 , f 3 = 1 1 , f 4 = 1 . Show that f 1 , f 2 , f 3 , f 4 are linearly independent. Name and SID: 4 3. Let S = { (1 , 2 , 3) , (2 , 3 , 4) , (3 , 4 , 5) } . Find the dimension and a basis for span ( S ). Name and SID: 5 4. Define a function T : R 2 → M 2 × 2 ( R ) as T ( a 1 , a 2 ) = ± a 1 a 2 2 a 2 a 1 ² . (a) Show that T is a linear transformation. (b) Find bases for its range and null spaces. Name and SID: 6 5. Find linear transformations U , T : R 2 → R 2 such that UT = T (the zero transformation) but TU 6 = T ....
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