Question1, Determine the values of a for which the system has no solution, exactly 1 solution or infinitely many solutions: x + 2y + z = 2 2x 2y + 3z…

Question1, Determine the values of a for which the system has no solution, exactly 1 solution or infinitely many solutions:

x + 2y + z = 2

2x − 2y + 3z = 1

x + 2y − (a^2 − 3)z = a

Question 2 :Let u1 = (1, 1, 0), u2 = (0, 1, 1), u3 = (1, 0, 1). Find scalars c1, c2, c3 such that c1u1 + c2u2 + c3u3 = (1, 0, 0).

Question 3 :Find the standard matrices for the following 2 linear operators on R 2 : (a) a reflection about the line y = x. (b) a rotation counterclockwise of 30◦ .

Question 4: Let A = −14 12 −20 17 . Find an invertible matrix P and a diagonal matrix D such that D = P −1AP.

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