Mathematics 244, section 1 – Linear Algebra Exam 1 – February 16, 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 consistent?

( \begin{matrix}−1 & 2 & 0 & c & | & 1 \\ 0 & 1 & 2 & 0 & | & −1 \\ 2 & −4 & 0 & 8 & | & 5 \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 independent. B) (10) For which value of a ∈ R is

S = \{( \begin{matrix}3 \\ 2 \\ 1 \end{matrix} ) , ( \begin{matrix}−1 \\ 1 \\ 0 \end{matrix} ) , ( \begin{matrix}1 \\ 0 \\ a \end{matrix} )\}

a linearly dependent subset of R^{3}?

C) (15) Let b = ( \begin{matrix}3 \\ 3 \\ 4 \end{matrix} ). Using your value of a from part B, is b ∈ Span(S)? Why or why

not?

III. Let T : R^{3} → R^{3} be the mapping defined by

T (x) = ( \begin{matrix}1 & 2 & 1 \\ −1 & 4 & 0 \\ 1 & 0 & 1 \end{matrix} ) x

A) (10) Show that T is a linear mapping by verifying the properties in the definition. B) (10) Is T an onto 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, then the set of solutions

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

(T_{A}(x) = Ax for x ∈ R^{n}). Then the span of the set of columns of A is R^{m}.