Mathematics 244, section 1 – Linear Algebra Makeup Exam 1 – February 21, 2007

Directions

Do all work in the blue exam booklet. There are 100 points.

I. A) (10) For which value(s) of c ∈ R is the system of linear equations represented by the

following augmented matrix inconsistent?

( \begin{matrix}−1 & 0 & 3 & c & | & −1 \\ 0 & 1 & 2 & 0 & | & 0 \\ 3 & 0 & −9 & 12 & | & 2 \end{matrix} )

B) (10) Give a parametrization of the set of solutions for a value of c (your choice) that makes the system consistent.

II. A) (5) Define: The set S of vectors in R^{n} is linearly dependent. B) (10) For which value(s) of a ∈ R is

S = \{( \begin{matrix}1 \\ 3 \\ 1 \end{matrix} ) , ( \begin{matrix}2 \\ 4 \\ 1 \end{matrix} ) , ( \begin{matrix}−1 \\ a \\ 1 \end{matrix} )\}

a linearly independent subset of R^{3}?

C) (15) Let b = ( \begin{matrix}3 \\ 3 \\ 4 \end{matrix} ). Using a = 1 in part B, is b ∈ Span(S)? Why or why not?

III. Let T : R^{3} → R^{3} be the linear mapping with T (e_{1}) = e_{1} + e_{3}, T (e_{2}) = −e_{2} + 2e_{3}, T (e_{3}) = e_{1} + e_{2}. A) (10) What is T (4e_{1} + 2e_{2} − e_{3})? B) (10) Is T a one-to-one mapping? Why or why not?

IV. (True-False) Determine whether each of the following statements is true or false. For the ones that are true, give short proofs; for those that are false, give counterexamples.

A) (10) If the homogeneous system Ax = 0 has two free variables, and the inhomogeneous

system Ax = b is consistent, then the set of solutions of Ax = b is a plane. B) (10) If x ∉ S and S ∪ {x} is linearly dependent, then x ∈ Span(S). C) (10) If A is an m×n matrix that defines an onto mapping T_{A} : R^{n} → R^{m} (T_{A}(x) = Ax

for x ∈ R^{n}), then m ≥ n.