APPM 3310: Matrix Methods — Exam #1 — Summer 2012
On the front of your bluebook write (1) your name, (2) TEST 1/3310, (3) SUMMER 2012 and a grading
table with 3 problems and a total. A correct answer with no supporting work may receive no credit while
an incorrect answer with some correct work may receive partial credit. Start each problem on a new
page. No electronic devices of any kind (e.g. cell phones, calculators, etc.) are permitted. A one-page sheet
of notes is allowed. SHOW ALL WORK. JUSTIFY ALL YOUR ANSWERS.
Problem 1: (30 pts) Consider the matrix A=
1−231
1−1 0 −1
2−1−3 3
.
(a) (6 pts) Find a decomposition A=LR where Lis special lower triangular and Ris in row echelon
form (Write down both Land R.) What is the rank of the matrix A?
(b) (6 pts) Is the set
1
1
2
,
3
0
−3
,
1
−1
3
a basis for the range of A? Why or why not?
(c) (6 pts) Find a basis for ker(A).
(d) (6 pts) Now given that b= [4,−1,0]Tfind the general solution of the system Ax=bgiven above.
(e) (6 pts) Consider the block matrix defined as B=
A A A
A A A
A A A
where Ais the same matrix given
above. Write the block matrix Bin REF (do not write out the 9×12 matrix, instead write it in block form
using the fact that R is the REF of A.) What is the rank of B? Justify your answer.
Problem 2: (35 pts) Let Vbe the set of vector valued functions in R2with components from the set of
polynomials of degree less than or equal to 2, that is
V≡ f(t)
g(t)
f(t), g(t)∈ P(2)
(a) (5 pts) Show that the elements of Vare closed under standard vector addition. Justify your answer.
(b) (5 pts) Show that the elements of Vare closed under standard scalar multiplication of vectors. Justify
your answer.
(c) (5 pts) Is R2a subset of V? Justify your answer.
(d) (10 pts) Assuming Vis a vector space with real scalars, write down a basis for V, justify your answer.
What is the dimension of V?
(e) (5 pts) Write down the coordinates of v=t
6+9t.
(f) (5 pts) Is v=t
6+9tin the span of the vectors v1=7
tand v2=t
π? Justify your answer.
Problem 3: (35 pts) Short answer. Each of the questions below are unrelated. Justify your answers.
(a) Define a norm in R2by kxk ≡ √xTx, show that if xTy= 0 then kx+yk2=kxk2+kyk2.
(b) Does hu,vi ≡ u1v1+ 2u2v3+ 3u3v2define a norm in R3? Why or why not?
(c) Always True or False: For any n×nmatrix A, det(ATA) = det(A2) (Justify your answer.)
(d) Always True or False: For an n×mmatrix A, if rank(A) = nthen ker(A) = {~
0}(Justify)
(e) If Fis 2 ×2 and you add row 1 of Fto row 2 of Fto get B, how do you find B−1from F−1?
(f) If Mis the inverse of F2, show that F M is the inverse of F.
(g) If Band Fare m×mand if Bis not invertible then is FB invertible?
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