318midterm 1.pdf

# 318midterm 1.pdf - Math 318 Midterm 1 Feb 17 2017 Read the...

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Math 318 Midterm 1 Feb. 17, 2017 Read the questions carefully and answer them in the spaces provided. Show necessary work for your answer. During your reasoning, you may use any theorems from the textbook or the lecture. But you need to state what the theorems said clearly. Name 1. Let a R n be a vector. Define V = { x R n : x · a = 0 } . Show that V is a subspace of R n . 2. Find a nonzero vector in R 3 which is perpendicular to both u = 2 3 0 and v = 1 0 1 1

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3. Let a = 1 1 and b = 3 4 be two vectors in R 2 . Find the projection of a on b . 4. Let T : R 2 R 2 be the linear transformation defined by rotating the plane π/ 2 counterclockwise. (a) Find the standard matrix [ T ] of T . (b) Show that T is invertible. Page 2
5. Suppose A is an invertible n × n matrix and x R n satisfies A - 1 x = 9 x . Calculate A 2 x . 6. Let the curve C be an ellipse in

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• Spring '14
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• Topology, Metric space, Topological space, General topology

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