Math 518 Spring 2002 NAME:
Do 5 of the following 8 problems
1. Given the matrix A=
11 036
22 017
−1 1 −201
41 306
with reduced echelon form
10 101
0 1 −102
00 011
00 000
.
(a) Find a basis of the null space and a basis for the range of the linear transformation
T(v) = Av.
(b) Is the linear transformation Tone-to-one? onto?
2. Let S={v1, v2, ..., vn}be a set of vectors in a vector space V. Show that Sis lin-
early independent if and only if every vector in the span of Scan be represented uniquely as
a linear combination of the elements in S.
3. Let Vbe a vector space, Wa subspace of V, and T∈ L(V , V ). Define the subset
T−1(W) = {v∈V|T(v)∈W}. Show that T−1(W) is a subspace of V.
4. Let P4(R) denote the space of all polynomials in xof degree less than or equal to 4
with coefficients in R. Define the linear map D:P4(R)→P4(R) as D(p(x)) = p0(x). Find
all eigenvalues of Dthen choose a basis Bof generalized eigenvectors and find the matrix of D
corresponding to B.
5. Let Vbe a finite dimensional vector space and T∈ L(V, V ). Suppose that range(T) =
range(T2). Prove that V= range(T)⊕null(T).
6. (a) Prove or disprove that the following is a linear transformation:
T:C∞→ C∞with T(f) = x·f0.
(b) Find the null space of T.
7. Define S:P4(R)→P4(R) as S(p(x)) = p(x−a), with a∈R. Find the matrix of
Swith respect to the standard basis {1, x, x2, x3, x4}.
8. Let Vbe a finite dimensional vector space, T∈ L(V), and v∈V. Let pdenote the
monic polynomial of smallest degree such that p(T)(v) = 0. Prove that pdivides the minimal
polynomial of T.
