Mathematics 244, section 1 – Linear Algebra Sample Exam 1 – February 9, 2007

Directions

There are 100 points.

I. Let S be a subset of R^{n}. A) (5) Give the definition of Span(S), the span of S. B) (15) Show that S is linearly dependent if and only if there exists some x ∈ S which

satisfies x ∈ Span(S \ {x}).

II. (15) Is

\{( \begin{matrix}1 \\ 1 \\ 0 \end{matrix} ) , ( \begin{matrix}−1 \\ 0 \\ 2 \end{matrix} ) , ( \begin{matrix}0 \\ 2 \\ 1 \end{matrix} )\}

a linearly independent subset of R^{3}? Why or why not?

III. (15) Put the augmented matrix of this system of linear equations into row-reduced echelon form and find a parametrization for the set of solutions:

\begin{aligned}2x_{1} & & & + & x_{3} & − & x_{4} & = & 0 \\ x_{1} & + & 3x_{2} & & & + & 2x_{4} & = & 1 \\ & & x_{2} & + & x_{3} & + & x_{4} & = & −1 \end{aligned} .

IV. A) (5) Let T : R^{3} → R^{3} be the linear mapping with T (e_{1}) = 5e_{1} + e_{3}, T (e_{2}) = 2e_{2} − e_{3},

T (e_{3}) = e_{1} − e_{2}. What is the matrix A such that T = T_{A}? B) (15) Is the matrix A you found in part A invertible? If so, find A^{−1}. If not, give a

reason why you know it is not invertible.

V. (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) Let A be an n × n matrix. If the system of linear equations Ax = b has just one

solution x for some b ∈ R^{n}, then it has just one solution for all b. B) (10) If W = Span({v, w}) then for any scalar a ≠ 0, W = Span({av + w, w}) also. C) (10) If a system of linear equations has more variables than unknowns, there is auto-

matically a nonzero solution.