Mathematics 244, section 1 – Linear Algebra Exam 2 – March 23, 2007

Directions

Do all work in the blue exam booklet. There are 100 regular points and 5 Extra Credit points (see III C below).

I. (15) Let V be a vector space and let S be a finite subset of V with Span(S) = V. If T is a linearly independent subset of V, show that |T | ≤ |S|. (Here |S| denotes the number of elements in S, and similarly for |T |.)

II. All parts of this question refer to the matrix

A = ( \begin{matrix}1 & −2 & 0 & s & 2 \\ −2 & −1 & −1 & 2s & −1 \\ 1 & 3 & 1 & s^{2} & −1 \end{matrix} )

A) (15) For which value(s) of the scalar s does A satisfy dim Nul(A) = 2? B) (10) Say s = 1. Give bases of Col(A) and Nul(A).

III. (10) Use Cramer’s Rule to solve the system

\begin{aligned}3x_{1} + 2x_{2} & = 9 \\ −8x_{1} + 3x_{2} & = −12 \end{aligned}

IV. Let V = M_{2×2}(R), the vector space of all 2×2 matrices with real entries. Let W = R^{2}.

A) (10) Is T : V → W defined by T (A) = ( \begin{matrix}det(A) \\ det(A^{2}) \end{matrix} ) a linear mapping? Why or why

not? B) (10) Show that

H = \{A = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ V : a + b = c + d\}

is a vector subspace of V. C) (5 Extra Credit) Find a basis for H from part B and determine its dimension.

V. (True-False) For each true statement, give a short proof. For each false statement, give a counterexample. (Note: As always, “true” means “true in every case.” Partial credit will be given for definitions or theorems that apply even if you don’t find a complete solution.)

A) (10) Let A be a 4 × 4 matrix in which R_{1} + 2R_{2} = R_{3} + R_{4} (R_{i} is row i of the matrix).

Then det(A) = 0. B) (10) There exist linear mappings T : R^{2} → R^{2} with standard matrices A satisfying

Nul(A) = Col(A). C) (10) Let A, B, C be n×n matrices. If det(A^{t}BC^{5}) = 23, then A, B, C are all invertible

matrices.