# exam5 - Name M340L Final Exam Problem 1 Find all...

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Name: M340L Final Exam May 13, 1995 Problem 1: Find all solutions (if any) to the system of equations. Express your answer in vector parametric form. x 1 + 2 x 3 + 3 x 4 =6 2 x 1 + 2 x 2 + x 3 - 3 x 4 =2 4 x 1 + 2 x 2 + 5 x 3 + 3 x 4 =14 Problem 2: T is a linear transformation from IR 3 to IR 4 defined by T x 1 x 2 x 3 = x 3 x 1 - x 2 x 2 x 1 + x 2 . a) Find the matrix of this linear transformation. b) Is T 1-1? If not, find a nonzero vector x such that T ( x ) = 0. c) Is T onto? If not, find a nonzero vector y such that y is not in the range of T . Problem 3: a) Compute the determinant of the matrix A = 4 8 8 8 5 0 1 0 0 0 6 8 8 8 7 0 8 8 3 0 0 8 2 0 0 b) Is A invertible? Why or why not? c) What is the rank of A ? Problem 4: Let E = { 1 ,t,t 2 } be the standard basis for IP 2 . Let B = { 1 + t + t 2 , 1 + 2 t + 3 t 2 , 1 + 4 t + 9 t 2 } be another basis. Let T : IP 2 IP 2 be the linear transformation T ( p ( t )) = p ( t ) + t ( dp ( t ) /dt ). a) Find the matrix of T relative to the standard basis E . Call this matrix A .

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