Name: AMAT 220: Linear Algebra Practice Exam March, 2021 Show all work for each problem in the space provided. If you run out of room for an answer, continue on the back of the page. You may NOT use a...

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Name: AMAT 220: Linear Algebra Practice Exam March, 2021 Show all work for each problem in the space provided. If you run out of room for an answer, continue on the back of the page. You may NOT use a calculator Question Points Bonus Points Score 1 0 0 2 0 0 3 0 0 4 0 0 5 0 0 6 0 0 7 0 0 8 0 0 9 0 0 Total: 0 0 1. Define what both a linear transformation is and the span of a set of vectors. 2. Find the general solution of the linear system 3x1 − 4x2 + 2x3 = 0 (1) −9x1 + 12x2 − 6x3 = 0 (2) −6x1 + 8x2 − 4x3 = 0 (3) OR x1 − 7x2 + 6x4 = 5 (4) x3 − 2x4 = −3 (5) −x1 + 7x2 − 4x3 + 2x4 = 7 (6) by row reducing the corresponding augmented matrix and interpreting your result in terms of the corresponding linear system. 3. Show by computation whether the given vector b ∈ Span{a1, a2, . . . , an}, where ai are vectors. Specifically, show whether b =  −9 −7 −15  is contained in the spanning set Span{a1 =  2 1 4  a2 =  1 −1 3  a3 =  3 2 5 } OR b =  9 2 7  is contained in the spanning set Span{a1 =  1 2 −1  a2 =  6 4 2 } 4. Describe in parametric form all solutions of the system Ax = 0 for the given A. A = 2 1 3 1 2 0  OR A = 3 1 1 1 5 −1 1 1  5. Determine if the given set of vectors are linearly dependent: S = {a1 =  −2 0 1  , a2 =  3 2 5  , a3 =  6 −1 1  , a4 =  7 0 2 } OR S = {a1 =  8 −1 3  , a2 =  4 0 1 } 6. Compute the standard matrix A associated to the given linear transformation T : Rn → Rm, where T (x1, x2) = (x1 + 2x2, 3x1 − x2) OR T (x1, x2, x3) = (2x1 − x2 + x3, x2 − 4x3) 7. Compute the matrix product AB for the given A and B. A = 1 2 4 2 6 0  and B =  4 1 4 3 0 −1 3 1 2 7 5 2  OR A =  6 1 3 −1 1 2 4 1 3  and B =  3 0 −1 2 1 1  8. Compute the inverse of the given matrix A by the algorithm for computing a matrix’s inverse; that is, row reduce the matrix of A augmented by I. A =  1 2 3 2 5 3 1 0 8  OR A =  3 4 −1 1 0 3 2 5 −4  9. Use your answer from the question 8. to find a solution x to the linear system x1 + 2x2 + 3x3 = 7 (7) 2x1 + 5x2 + 3x3 = 5 (8) x1 + 8x3 = 8 (9) RESPECTIVELY 3x1 + 4x2 − x3 = 7 (10) x1 + 3x3 = 5 (11) 2x1 + 5x2 − 4x3 = 8 (12) Respectively means apply the inverse of the first answer in 8 to solve the first problem in this question and to apply the second answer in 8 to solve the second problem in this question.