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exam2math3042009solns

exam2math3042009solns - Exam 2 Math 304 2009 1 Dene the...

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Exam 2 Math 304, 2009 1. Define the following terms, or answer true-false, as appropriate: (The carefully worded definitions can be looked up in your book. I’ll provide one such, and for the rest, I’ll limit the definition to a sketch that includes the essential core information. (a) a linear transformation from a vector space U to a vector space V . A linear transformation L from U to V is a function L from U to V such that for all u 1 and u 2 in U , and all scalars α , L ( u 1 + u 2 ) = Lu 1 + Lu 2 , and L ( αu 1 ) = αL ( u 1 ). Short version: L ( αu 1 + βu 2 ) = αLu 1 + βLu 2 . (b) the kernel of a linear operator on U . { u : Lu = 0 } . (c) an inner product on a vector space U , when the field is R . ( u, v ) = ( v, u ) , ( u, v + w ) = ( u, v ) + ( u, w ) , and ( αu, v ) = α ( u, v ) . (d) an orthogonal matrix . Q T = Q - 1 , OR, Q is a square matrix whose columns form an orthonormal set. (e) the orthogonal complement of a vector subspace V of an inner product space U . { u U such that u v v V } . (f) an orthonormal set of vectors in an inner product space. The vectors are mutually orthogonal and all have length 1. (g) a unit vector in an inner product space. A vector v so that ( v, v ) = 1. Also, but not as good a definition because it ducks the question of how length is measured, a vector v with bardbl v bardbl = 1. (h) The product of two orthogonal matrices is again an orthogonal ma- trix. True. Reasons weren’t required, but here’s the reason: if P and Q are orthogonal n × n matrices, then Q T Q = I and P T P = I .
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