Matrix Overview

# Matrix Overview - 1 P roblem Set 1 1 What are the axioms...

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1 Problem Set 1 1. What are the axioms that define a vector space? 2. Prove that any collection of vectors in <n containing the zero vector cannot be linearly independent. 3. Let B be a pxn matrix of real numbers. Prove that the set of solutions to Bx = 0 is closed under linear combination and hence is a vector subspace in R n . 4. Let S be the triangle in R 2 with vertices at (0,0), (0,1) and (1,0) Consider the mapping f : R 2 =>R 2 given by f1(x; y) = x + y and f2(x; y) = x - y. Sketch f(S). 5. Find an ortho-normal basis for the vector space spanned by (8;-5; 2), (1; 2; 4), and (3;-1; 2). 6. Let PX = X(X’X) -1 X’ be the nxn projection matrix onto the vector space spanned by the columns of the n x k matrix X. Let X1 be any matrix composed of a linearly independent subset of the columns of X such that rank(X) = rank(X1). Prove that PX = X(X’X) -1 X’ . Why is (PX - I) called the annihilator matrix? 7. Suppose X is an nxk matrix, and Y is the matrix obtained by taking the first k - 1 columns of X. Prove that (X’X) -1 is positive semi-definite. Let A be the matrix obtained by deleting the kth row and column of (X’X). Is A-Y’Y necessarily positive semi-definite? (This problem is inspired by a practical problem that emerges with the Hausman specification test). 1) If the following axioms are satisfied by all objects u , v , w in V and all scalars k and l, then V is a vector space and the objects in V are vectors . (i) If u and v are objects in V, then

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Matrix Overview - 1 P roblem Set 1 1 What are the axioms...

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