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MT1sol - Stony Brook University Mathematics Department...

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Stony Brook University Introduction to Linear Algebra Mathematics Department MAT 211 Julia Viro March 10, 2009 Solutions for Midterm I Problem 1. Solve the system using the Gauss-Jordan elimination and verify your answer x 1 + 2 x 3 - x 4 =1 x 2 + 2 x 4 = - 1 x 1 - x 2 + 2 x 3 - 3 x 4 =2 . Solution. Elementary row transformations of the augmented matrix of the system give rise to the reduced row-echelon form: 1 0 2 - 1 1 0 1 0 2 - 1 1 - 1 2 - 3 2 R 3 - R 1 1 0 2 - 1 1 0 1 0 2 - 1 0 - 1 0 - 2 1 R 3+ R 2 1 0 2 - 1 1 0 1 0 2 - 1 0 0 0 0 0 . Obviously, the rank of the matrix is 2. The number of free variables is 4 - 2 = 2 (the number of unknowns minus the rank). We write down the solution starting from the back: x 4 = t (choose x 4 as a free variable), x 3 = s (choose x 3 as a free variable), x 2 = - 1 - 2 t (from the second row in the rref), x 1 = 1 + t - 2 s (from the first row in the rref). So the solution is ( x 1 , x 2 , x 3 , x 4 ) = (1 + t - 2 s, - 1 - 2 t, s, t ) = (1 , - 1 , 0 , 0) + t (1 , - 2 , 0 , 1) + s ( - 2 , 0 , 1 , 0) , where t and s are arbitrary real numbers. Geometrically, the solution is a plane in R 4 . To verify the solution, we substitute it into the three equations of the system: 1 + t - 2 s + 2 s - t = 1 - 1 - 2 t + 2 t = - 1 1 + t - 2 s + 1 + 2 t + 2 s - 3 t = 2 . Since all the equations are satisfied, our solution is correct. Answer: ( x 1 , x 2 , x 3 , x 4 ) = (1 + t - 2 s, - 1 - 2 t, s, t ) , t, s R .
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2 Problem 2. Let T : R 2 R 2 be an orthogonal projection onto the line x - 2 y = 0 followed by a counterclockwise rotation by 45 . Find a matrix of T (with respect to the standard basis). Describe geometrically and show on a picture the kernel and the image of T . Is T invertible?
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