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final_exam - to show that rank A T A = rank A and null A T...

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MA 35100 GOINS FINAL EXAM This exam is to be done in 2 hours in one continuous sitting. Collaboration is not allowed, but you may use your own personal notes; the lecture notes posted on the course web site; homework solutions posted on the course web site; and Bretscher’s Linear Algebra with Applications . Write your answers on a separate sheet of paper. Provide all details of your work : either give precise references for or proofs of any statements you claim. Problem 1. Use Gauss-Jordan elimination to solve the linear system x - 2 y + 2 z = 0 2 x - 5 y + 6 z = - 1 4 x - 13 y + 18 z = - 5 Problem 2. Consider a line L in the coordinate plane, running through the origin. Denote proj L : R 2 → R 2 as the orthogonal projection onto L , and T : R 2 → R 2 as the linear transformation T ( ~x ) = 0 - 1 1 0 ~x. What is the image of the composition T ◦ proj L as a subspace of R 2 ? Give a geometric interpretation. Problem 3. Let A be an n × m matrix, and A T denote its transpose. Use the Rank-Nullity Theorem
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Unformatted text preview: to show that rank( A T A ) = rank( A ) and null( A T A ) = null( A ) . Problem 4. Consider the linear transformation T : R 3 → R 3 defined by T ( ~x ) = M ~x in terms of the 3 × 3 matrix M = 2 9 13 3 3 2 6 20 11 . a. Compute the QR-factorization of M . b. Find an orthonormal basis for the image of T . Problem 5. Let V and W be linear spaces with inner products h·i V and h·i W , respectively. A linear transformation T : V → W is said to be an isometry if it preserves lengths i.e., || T ( f ) || W = || f || V for all f ∈ V . a. Show that every isometry is a monomorphism i.e., if T is an isometry then ker( T ) = { } . b. Say V = W = R n , and that h·i V = h·i W is the dot product. Show that T is an isometry if and only if it is orthogonal....
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