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Unformatted text preview: MATH. 20F SAMPLE MIDTERM 1 SOLUTIONS You have 50 minutes for this exam. Please write legibly and show all working. No calculators are allowed. Write your name and ID number. Name: ID Number: (1) Let A = 1 2 1 1 a 2 1 3 . (i) (8 points) Find all a s such that the homogeneous system Ax = 0 has non-trivial solutions. After performing Gaussian elimination on A , one obtains the row echelon form 1 2 1 0 3 1 0 0 a 5 . It follows that there is a free variable if and only if a = 5. Hence, Ax = 0 has infinitely many solutions if and only if a = 5. (ii) (7 points) For those values of a from (i), find the general solution of Ax = 0. Taking a = 5 in the row echelon form from (i), we see that x 3 is a free variable. Setting x 3 = , we see that x 2 = 1 3 and x 1 = 5 3 . So the general solution is x = ( 5 3 , 1 3 , 1) t . 1 2 MATH. 20F SAMPLE MIDTERM 1 SOLUTIONS (2i) (5 points) Let T : R n R m be a linear transformation. If x and...
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