235_tt1_s10_soln

235_tt1_s10_soln - Math 235 Term Test 1 Solutions 1. Short...

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Unformatted text preview: Math 235 Term Test 1 Solutions 1. Short Answer Problems [2] a) By considering the dimension of the range or null space, determine the rank and the nullity of the linear mapping T : P 2 R 2 , where T ( p ( x )) = p (0) p (1) . Solution: Range( T ) = R 2 since T (1- x ) = 1 and T ( x ) = 1 . Thus, rank( T ) = 2 and nullity( T ) = dim( P 2 )- rank( T ) = 3- 2 = 1. [2] b) State the definition of an orthonormal set in an inner product space V . Solution: A set { ~v 1 ,...,~v n } of vectors in V is orthonormal when h ~v i ,~v i i = 1 for all i h ~v i ,~v j i = 0 for all i 6 = j . [1] c) Define the rank of a linear mapping L : V W . Solution: rank( L ) = dim(Range( L )). [1] d) Give the definition of an orthogonal matrix. Solution: An orthogonal matrix P is a matrix such that P T = P- 1 . [2] e) Determine if h p,q i = p (- 1) q (- 1) + p (1) q (1) is an inner product for P 2 . Solution: It is not since h x 2- 1 ,x 2- 1 i = 0 but x 2- 1 6 = ~ 0 in P 2 . 1 2 2. Let A = 3 1 4 2 3 5 2 7 3 4 2- 1 1 3 7 3 2 5 2 1 , then the RREF of A is R = 1 0 1 0 1 0 1 1 0- 2 0 0 0 1 1 0 0 0 0 . [2] a) Find rank( A ) and dim(Null( A )). Solution: rank( A ) = 3 nullity( A ) = (# of columns)- rank( A ) = 5- 3 = 2. [2] b) Find a basis for Row( A ). Solution: U = 1 1 1 , 1 1- 2 , 1 1 [2] c) Find a basis for Null( A )....
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235_tt1_s10_soln - Math 235 Term Test 1 Solutions 1. Short...

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