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Unformatted text preview: Math 311.503/505  Practice for Midterm 2 • 1. Answer TRUE or FALSE. – i) Let k · k be a norm in R n . Then for every x,y ∈ R n one has that k x y k 2 + k x + y k 2 = 2 k x k 2 + 2 k x k 2 . – ii) Let A be a 5 × 3 matrix with rank( A ) = 2. Then dim N ( A ) = 1. – iii) Let S be a subspace of R n . Then the only element in the intersection of S and S ⊥ is the zero vector. – iv) Let A be an m × n matrix. Then the number of linearly independent columns is equal to the number of linearly independent rows. – v) Let A,B,C be n × n matrices. Then tr( ABC ) = tr( CAB ). – vi) The functions 1, x 2 are orthogonal in the space C ([ 1 , 1]) with inner prod uct h f,g i := R 1 1 f ( x ) g ( x ) dx . – viii) If x,y nonzero vectors in R 2 with h x,y i = 0 have the property that span { x 1 ,x 2 } = R 2 . • 2. Determine whether the following are linear transformations: 1. L : R n × n → R n × n , L ( A ) = L ( A T ) , 2. L : C ([0 , 1]) → R 2 , L ( f ) := R 1 f ( x ) dx,...
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This note was uploaded on 11/10/2011 for the course MATH 311 taught by Professor Anshelvich during the Spring '08 term at Texas A&M.
 Spring '08
 Anshelvich
 Math

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