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4 - NIE2009PraticeFinal(NoSolutions)

# 4 - NIE2009PraticeFinal(NoSolutions) - 2 9 x 3 =-6 Show all...

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Practice Final of Math 20F * Jiawang Nie December 2, 2009 Notes: 1)For computational problems, no credit will be given for unsupported answers gotten directly from a calculator. 2)for proof problems, no credit will be given for wrong reasons. 1. Find all the eigenvalues and associated eigenvectors of the matrix - 1 4 - 2 - 3 4 0 - 3 1 3 . Show all the computational steps of how you get the answer. 2. Find a diagonalization A = PDP - 1 for the matrix A = 2 3 1 4 1 1 0 0 1 . Here P R 3 × 3 is invertible and D R 3 × 3 is diagonal. Show all the computational steps of how you get the answer. 3. Determine whether the following matrix 0 1 2 1 0 3 4 - 3 8 . is invertible or not. If so, ﬁnd the inverse; if no, give reasons. 4. Describe the solution set of the following linear system in parametric vector form x 1 + 3 x 2 - 5 x 3 = 4 x 1 + 4 x 2 - 8 x 3 = 7 - 3 x 1 - 7 x

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Unformatted text preview: 2 + 9 x 3 =-6 Show all the computational steps of how you get the answer. * The real exam might be totally diﬀerent !!! 1 5. Find the rank, the dimension and a basis of the column space of the matrix 1-4 9-7-1 2-4 1 5-6 10 7 . Show all the computational steps of how you get the answer. 6. Let v 1 ,v 2 ,v 3 ,v 4 ∈ R 4 . Suppose v 1 ,v 2 ,v 3 are linearly dependent. Show that v 1 ,v 2 ,v 3 ,v 4 must also be linearly dependent. 7. Let A,B ∈ R n × n be such that AB is invertible. Show that A and B are both invertible. 8. Let A ∈ R n × n be such that A 2 = 0. Show that all the eigenvalues of A are zero. 9. Let u,v ∈ R n . Show that if k u + v k 2 = k u k 2 + k v k 2 then u and v are orthogonal. 2...
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4 - NIE2009PraticeFinal(NoSolutions) - 2 9 x 3 =-6 Show all...

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