MATH 110, mock midterm test.
Name
Student ID #
All the necessary work to justify an answer and all the necessary steps of a proof must be
shown clearly to obtain full credit. Partial credit may be given but only for significant
progress towards a solution. Show all relevant work in logical sequence and indicate all
answers clearly. Cross out all work you do not wish considered. Books and notes are allowed.
Calculators, computers, cell phones, pagers and similar devices are not allowed during the
test.
1. Consider the vector space P(IR) and the subsets Vconsisting of those vectors (polyno-
mials) ffor which:
(a) fhas degree 3,
(b) 2f(0) = f(1),
(c) f(t)≥0 whenever t≥0,
(d) f(t) = f(1 −t) for all t.
In which of these cases is Va subspace of P(IR)?
2.
Let A=0 2 3
−1 3 −2, B =
−2 0
6 4
−4 6
, v =123, w =
0
−1
2
.
Do the products Aw,Btvt,vAw exist? Evaluate those that do. Is the set {A, B t}linearly
independent?
3.
Let A=0 2
2−2, β ={−1
1,1
1}.
Find the representation [LA]β, the dual basis β∗, and the matrix [(LA)t]β∗.
4. Let
A:Pn(IR) →Pn(IR) : (Af )(t) := f(t+ 1).
Prove that
A=I+D
1! +D2
2! +· · · +Dn
n!,
1
