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final_sample

# final_sample - plane related to each other Problem 3 Let A...

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MA 35100 GOINS SAMPLE 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 + y + z = 3 2 x + 3 y + 4 z = 11 4 x + 9 y + 16 z = 41 Problem 2. Fix two scalars a and b such that a 2 + b 2 = 1. Consider the identity a b b - a = a - b b a 1 0 0 - 1 . What is the geometric interpretation of this composition? Draw pictures giving a geometric interpretation of each transformation. How are reflections and rotations in the coordinate
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Unformatted text preview: plane related to each other? Problem 3. Let A be an n × m matrix. Assume that there exists an m × n matrix B such that B A = I n . Show that rank( A ) = m . Must A necessarily be invertible? Problem 4. Consider the linear transformation T : R 3 → R 3 deﬁned by T ( ~x ) = A~x in terms of the 3 × 3 matrix A = 1 7 2 3 6 3-6 2 6 2-3 . a. Show that T is an orthogonal transformation. b. Find the matrix of the inverse of T . Problem 5. Let V be a linear space with inner product h·i . Prove the Law of Cosines : Show that if θ is the angle between f,g ∈ V , then || f-g || 2 = || f || 2 + || g || 2-2 cos θ || f |||| g || ....
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