Mathematics 244, section 1 – Linear Algebra Exam 1 – February 20, 2004

Directions

Do all work in the blue exam booklet. There are 100 regular points and 10 Extra Credit points.

I. (15) Let V = P_{4}(R), and

W = {p ∈ V : p′(2) = 0 and p(0) = 0}

Is W a vector subspace of V? Why or why not?

II. Let S be a subset of a vector space V. 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}).

III. (15) Is {(1, 1, 0), (−1, 0, 2), (0, 2, 1)} a linearly independent subset of R^{3}? Why or why not?

IV. (20) Put this system of linear equations into 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}

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) If S is linearly dependent in V and S ⊂ T, then T is also linearly dependent. B) (10) If W is a subspace of V and 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.

Extra Credit: (10) In F (R), for each a ∈ R, we can define a function f_{a} by

f_{a}(x) = \{ \begin{matrix}1 & if x = a \\ 0 & otherwise \end{matrix}

Show that S = {f_{a} : a ∈ R} is a linearly independent subset of F (R). (For those who have taken Principles of Analysis, note that this shows there is an uncountably infinite linearly independent subset of F(R). It can be shown that this implies that no basis can be even countably infinite.)