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exam1sol - MATH 115A - Lecture 2 - Winter 2011 Midterm 1 -...

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Unformatted text preview: MATH 115A - Lecture 2 - Winter 2011 Midterm 1 - January 25, 2011 NAME: STUDENT ID #: This is a closed-book and closed-note examination. Calculators are not allowed. Please show all your work. Use only the paper provided. You may write on the back if you need more space, but clearly indicate this on the front. There are 5 problems for a total of 100 points. POINTS: 1. 2. 3. 4. 5. 1 2 1. (20 points) Let P 2 be the R-vector space of polynomials f ( x ) = a + bx + cx 2 of degree at most two. Is the set S = { 1 + x, 2- x 2 ,x + x 2 } a basis for P 2 ? If so, prove it. If not, disprove it. Solution: As the dimension of P 2 is 3, it suffices by the replacement theorem to prove either that S is a generating set or that it is linearly independent. Lets do the former. To show that S generates the vector space, it is enough to prove that a basis of the space is in Span( S ). Now we observe that 1 = (2- x 2 )+( x + x 2 )- (1+ x ), so 1 is in the span. Therefore, x = (1+ x )- 1 is as well, and hence so is...
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exam1sol - MATH 115A - Lecture 2 - Winter 2011 Midterm 1 -...

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