Mathematics 244, section 1 – Linear Algebra Midterm Exam 2 – March 26, 2004

Directions

Do all work in the blue exam booklet. There are 100 regular points in questions I - III, and 10 Extra Credit points. Standard 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. Let T : P_{3}(R) → R^{3} be defined by

T (p(x)) = (p(1), p′(1), p′′(1)).

A) (10) Show that T is linear. B) (10) Show that α = {1, x + x^{2}, x^{2} + x^{3}, x^{3} + 1} is a basis for the vector space P_{3}(R)

(don’t work too hard!) C) (10) Find the matrix of T with respect to the bases α = {1, x + x^{2}, x^{2} + x^{3}, x^{3} + 1}

for P_{3}(R) and and β = the standard basis in R^{3}. D) (15) Let S : R^{3} → R^{3} be the linear mapping defined by

S(x_{1}, x_{2}, x_{3}) = (3x_{1} − x_{2}, x_{1} + x_{2} + x_{3}, x_{1}).

What is the matrix of ST : P_{3}(R) → R^{3}, with respect to the bases α and β as in part C?

II. (25) State and prove the Dimension Theorem for linear mappings.

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 : R^{3} → R^{5}, then there are some b ∈ R^{5} such that the equation T (x) = b

has no solutions x ∈ R^{3}. B) (10) If T : P (R) → P (R) is a linear mapping and T is not injective, then T is not

surjective either. C) (10) There exists a linear mapping T : R^{4} → R^{7} which has dim Ker(T) = 2, and

whose image contains the vectors

(1, 1, 1, 0, 0, 0, 0), (0, 0, 0, 1, 1, 1, 1), (1, 0, 1, 0, 1, 0, 1).

Extra Credit (10) We know one proof of the following statement via the process of reduction of a system of equations to echelon form: Every homogeneous system of m linear equations in n variables with n > m has a nontrivial solution. Give a different proof of this statement using the Dimension Theorem for linear mappings (problem II).