Math. 3013: Sample Test Problems Page 1
Show all your work. Circle your answers to each problem clearly. You may use
graphing calculators no more powerful than a TI-80+ series calculator. Point
values of each problem are indicated in brackets. The total is 100 points.
1. With respect to the picture below of three plane vectors x,uand v, carefully estimate
the scalars rand ssuch that x=ru+sv. Draw any lines you may need on the picture.
u
v
x
2. Evaluate the following vector and matrix calculations.
a) [1,2,3] + 3 [−1,0,2] b) −1−x
1x−1x−1x
−1−1
3. Determine all values of csuch that the vector [c, c, c3] is parallel to [1,1,4]. Prove your
answer.
4. Find the distance between the points [1,2,−1,4] and [1,−1,1,−2].
5. For vectors vand win Rn, prove that v−wand v+ware perpendicular if and only
if kvk=kwk.
6. Let Anbe the n×nmatrix with ij entry defined by aij =i−j.
(a) Explicitly write down the 4 ×4 case A4.
(b) Write down a simple relationship between Anand AT
n.
7. A transition matrix Tfor a Markov process with four states has the form
T=
+ + 0 0
+ + + 0
0 + + +
0 0 + +
where the +’s denote positive entries. Determine whether or not Tis a regular tran-
sition matrix. If it is not, explain which entries are 0 in higher powers Tk. If it is,
determine the smallest power Tkwhich has all positive entries.
(turn page)
