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Problem 1.[5] Let X1 = (1,-1,1,1), I2 = (2,-1,1,1),2 I3 =(-1,-1,1,1),x4=(1,-3,2,2),y1 = (-1,2,-1,-2),32 = (2,-2,1,3),y = (0,0,1,-2) be (column) vectors in C4. Let U = and let V = Span {Y1,Y2,Y3}. Find a basis for UnV. Justify your answer. You can use the fact that the reduced row echelon form of the matrix (x1 X2 X3 X4 y1 Y2 y3) is 103001-1 01-20001 00010-1 2 00001-23 Problem 2.[10] (a) Let V = M2(C), let A (:4) be a fixed matrix and let LA: A-> A be the linear map given by LA(X) = AX. Write the matrix form of LA in your favourite basis of V. (b) Let V = {A € M2(C) tr(A) = 0} (here tr stands for the trace, i.e., the sum of diagonal entries of a matrix). Let e let X (=ae+bf+ch for some fixed complex numbers a. Write the matrix form of the linear map Dx: V-> given by Dx(Y)=XY-YX relative to the basis e,h,f of V. Problem 3.[10] Let V be a vector space over a field F and let T: V -> V be a linear map. We say that a nonzero vector v E V is a generalized eigenvector for T corresponding to an eigenvalue AE F if for some positive integer k we have that (T - 1)*v = 0. (a) Let Vx be the set of all generalized eigenvectors corresponding to 1 together with the zero vector. Prove that Vx is a vector subspace of V. We call V1 the generalized eigenspace corresponding to A. (b) Let V1, Um € V be generalized eigenvectors corresponding to pairwise distinct eigen- values A1,.... Am. Prove that they are linearly independent. Problem 4.[10] (a) Find the eigenvalues and a basis for each eigenspace of of the matrix (b) Find a basis for the generalized eigenspace corresponding to 1 of the matrix A = 633 1-1-2 Problem 5.[5] Let M3(C). Let V be the subspace of C3 spanned by v = (1,1,1) and let U = C3/V. Verify that V is an invariant subspace for A and write the the matrix form of the induced linear map A: U -> U in your favourite basis of U.

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