Math 5310 Midterm 2 6 November 2015

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
(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!)