MATH311Midterm2Practice

MATH311Midterm2Practice - Math 311.503/505 Practice for...

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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 non-zero 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 0 f ( x ) dx, f (0)+ f (1) 2 T , 3. L : R 2 R 3 , L ( f ) := ( x 1 , x 1 + x 2 , x 1 x 2 ) T .
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