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5 - MCGOWNPraticeFinal(NoSolutions)

# 5 - MCGOWNPraticeFinal(NoSolutions) - Math 20F Practice...

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Math 20F Practice Final 1. Consider the system: x 1 + x 2 + 2 x 3 - 3 x 4 = - 5 2 x 1 + 2 x 2 + 5 x 3 - 4 x 4 = 6 3 x 1 + 3 x 2 + 8 x 3 - 5 x 4 = - 5 Determine the solution set to the system and write it in parametric form.

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2. Fix a positive integer n and let P n denote the set of all polynomials of degree less than or equal to n (with coefficients in R ). Let T : P n P n +1 be the transformation that sends a polynomial to its indefinite integral with C = 0; more precisely, ( Tf )( x ) = x 0 f ( t ) dt . (a) Prove that T is linear. (b) Describe the range of T ?
3. Let W denote the span of { v 1 , v 2 , v 3 } in R 3 , where v 1 = 1 - 1 0 , v 2 = 2 0 1 , v 3 = 1 3 2 . (a) What is the dimension of W ? (b) Find an orthogonal basis for W . (c) Let y = (1 , 1 , - 2). Compute the projection of y onto W .

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4. Find the LU factorization of the matrix A = 4 3 - 5 - 4 - 5 7 8 6 - 8 .
5. Consider the matrix A = 2 1 1 0 1 2 0 0 0 0 3 0 0 0 0 - 2 (a) Find a basis for each eigenspace of A .

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