practice-problems3_2012 - M ATH 2090P RACTICE P ROBLEMS N...

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MATH 2090–P RACTICE P ROBLEMS FOR T EST 3 N OVEMBER 2012 Use short, precise and complete English sentences to explain carefully your answers. Supply enough words so that the steps in your reasoning can be easily followed. (1) Let A = 1 - 1 0 0 2 1 3 - 1 - 1 - 2 - 3 1 3 0 3 - 1 . Find a basis for the nullspace of A . (2) Recall that the trace of a square matrix A , denoted Tr( A ), is the sum of the coefficients of A that are on the diagonal. Let S = { A M 2 ( R ) : Tr( A ) = 0 } . Find a basis for S . (3) Let B = { (1, - 1,0),(2,1,1),(1,1,1) } R 3 . (a) Show that B is a basis of R 3 (b) Let v = (3,0,1). Find the component vector [ v ] B of v relative to B . (4) Let B = { (2,1),(1,3) } and let C = { ( - 1,1),(2, - 3) } . It is easy to see that both B and C are bases of R 2 . Find the matrix P C B (the change-of-basis matrix from B to C ). (5) Let A = 1 2 1 1 1 2 2 0 1 2 0 2 . Find a basis for the column space of A . (6) Let A = 1 3 0 1 1 2 2 0 . Determine the nullity of A “by inspection” by appealing to the Rank-Nullity Theorem. Avoid computations. (7) Let T : P 2 P 2 be the linear transformation given by T (1 - t ) = 1 + t 2 , T (2 - t + 2 t 2 ) = 3 - t , T ( - 2 + t - t 2 ) = 2 + t

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