chuang_exam2

# chuang_exam2 - 2 Consider the(ordered bases α =(1,x,x 2...

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MATH 107: EXAM 2 No calculators allowed. Your work must be clearly written in order to receive credit. Time: 50 minutes (1) (10 pts) Let A = 3 0 0 - 9 - 1 5 - 1 - 1 3 (a) (9 pts) Compute the eigenvalues of A and a basis for each eigenspace. Clearly label which is which. (b) (1 pt) Is A diagonalizable? Why or why not? (2) (5 pts) Suppose matrix A has eigenvalues λ = 1 , 2 , 3 with algebraic multi- plicities 4 , 2 , 3 respectively. (a) (2 pts) How many possible Jordan forms are there? Be sure to show how you compute this number. (b) (3 pts) Find the precise Jordan form (up to permutation of Jordan blocks) if A satisﬁes the following data: k null( A - I ) k null( A - 2 I ) k null( A - 3 I ) k 1 2 1 2 2 3 3 (3) (25 pts) Let P k be real polynomials of degree k . Consider L : P 2 P 2 given by L ( f ) = f 00 + f 0 - 2 f where P 2 is the vector space of real polynomials of degree
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Unformatted text preview: 2. Consider the (ordered) bases α = (1 ,x,x 2 ) and β = ( x 2 + 1 ,x, 2 x 2 + x-1). (a) (5 pts) Compute [ L ] α α , the matrix representing L relative basis α . (b) (10 pts) Compute [ L ] β β , the matrix representing L relative basis β . (c) (8 pts) Compute P β α , the matrix changing α-coordinates to β-coordinates. (d) (2 pts) What are the kernel and image of L ? Clearly label which is which. (4) (10 pts) Let P k be real polynomials of degree ≤ k , and L : P 1 → P 2 be integration (without arbitrary constant). For example, L (1) = x . Let α = ( x + 1 ,x-1) and β = ( x 2 + 1 ,x + 1 , 2 x 2-1) be (ordered) bases for P 1 ,P 2 respectively. Find [ L ] β α . 1...
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