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Math 5310 Midterm 2 6 November 2015 1. Let V be a finite-dimensional vector space over a field F , and let T : V → V be a linear operator. (a) (2 pts) Let λ be an element of F . Define the λ-eigenspace of T (denoted Vλ in class). Also define the λ-generalized-eigenspace of T (denoted Vλgen in class). (b) (2 pts) Show that both Vλ and Vλgen are subspaces of V . (c) (2 pts) Show that both Vλ and Vλgen are T -stable. (Recall that a subspace W of V is said to be T -stable if T (W ) is contained in W .) 2. For each of the following matrices A, compute (i) the characteristic polynomial A; (ii) the Jordan canonical form of A; and (iii) a matrix P such that P −1 AP is equal to the Jordan canonical form of A. −1 4 (a) (4 pts) A = . −2 5 0 4 (b) (4 pts) A = . −1 4 3. In this question, consider C as a vector space over R. (a) (2 pts) What is the dimension of C as a real vector space? Exhibit a basis. √ (b) (3 pts) Multiplication by −1 defines a function I : C → C. Show that I is an R-linear transformation, and compute the matrix of I in the basis you exhibited in the previous part of the question. (c) (3 pts) Show that the matrix of I depends on your choice of basis, but that the matrix of I 2 is independent of the choice of basis. 4. Let V and W be vector spaces over a field F . Assume that V is finitedimensional. Let T : V → W be a homomorphism of vector spaces (i.e., a linear map). (a) (6 pts) Show that dim(im(T )) is finite and that dim V = dim (ker(T )) + dim (im(T )) . You may assume results proven in class prior to our discussion of this theorem; just be sure to state precisely what you are using. (b) (2 pts) Use the above dimension formula to show that if n is greater than m, then any homogeneous system of m linear equations in n unknowns has a non-zero solution. (In particular, this result can be proven without any appeal to Gaussian elimination!) 1