Mathematics 244, section 1 – Linear Algebra Retest Exam 2 – March 31, 2004

Directions

Do all work in the blue exam booklet. There are 100 regular points and 10 Extra Credit points. Notation:

• P_{n}(R) is the vector space of polynomials of degree ≤ n.

• P (R) is the vector space of all polynomials (no restriction on degree).

I. A) (15) Let V be a vector space with a finite spanning set S. If T is a linearly independent

subset of V, show that |T | ≤ |S|. B) (10) Use part A to deduce that if V be a vector space with a finite spanning set, then

all bases of V have the same number of elements.

II. Let T : R^{4} → P_{2}(R) be defined by

T (a_{1}, a_{2}, a_{3}, a_{4}) = (a_{1} + a_{2} − a_{4}) + (a_{2} − 3a_{3})x + (−a_{1} − 2a_{2} + 3a_{3} + a_{4})x^{2}.

A) (10) Show that T is linear. B) (15) Find the matrix of T with respect to α = the standard basis in R^{4} and β =

{1 + x, x, x^{2} − 1} in P_{2}(R). C) (10) Find a basis for, and the dimension of, Ker(T). D) (10) What is the dimension of Im(T)? Is T surjective?

III. (True-False) For each true statement, give a short proof. For each false statement, give a reason or a counterexample. A) (10) If T : V → W is linear, dim(V) = 5, dim(W) = 3, and S = {v_{1}, v_{2}, v_{3}} ⊂ Ker(T),

then S is linearly dependent. B) (10) Let β be the standard basis in R^{2}. If S = R_{2π/3} and T = R_{4π/3} are the counter-

clockwise rotations about the origin, then

[ST]_{β}^{β} = ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ) .

C) (10) If T : V → W is linear and dim Ker(T) = 0, then T is injective.

Extra Credit (10) Let T be the mapping from question I, and let S : P_{2}(R) → P_{2}(R) be the mapping defined by S(p(x)) = xp′(x). What is the matrix of ST with respect to α = standard basis in R^{4} and β as in question II part B in P_{2}(R)?