## Transcribed Text

[6]
1.
(a) Among all the points on the plane V in R4 spanned by the vectors
vi= (1,1,1,1) and U2 = (-1,0,1,2),
find the one that is closest to the vector y = (-5,3,2,6). Moreover, express
this closest vector as a linear combination of V1 and U2.
[4]
(b) Consider the data points
x
-1.0
1.0
2.0
y
-2.5
1.5
1.0
3.0
Use your results from part (a) to determine the straight line which, in
the vertical least square sense, best fits all the data points.
[5]
2.
(a) Find the eigenvalues of the matrix
A
=
and calculate their algebraic and geometric multiplicities. Moreover, deter-
mine all the eigenvectors of A. Is A diagonable? (Justify your answer!)
[5]
(b) Determine the constituent matrices of A and use these to find a formula for
An, for n = 1, 2,
[3]
3.
(a) Write the linear difference equation
5an+2 - 4an+1 - an
as a discrete linear system of the form Un+1 = Avn.
[7]
(b) Solve the difference equation when ao = 2, a1 1, and use your solution to
calculate lim an.
n-00
Page 2 of 4 pages
4. A certain country has two provinces P1 and P2. The yearly censes taken in the
country over the last 20 years shows that each year 10% of the residents of P1
move to P2, and that 15% of the residents of P2 move to P1.
3
(a) Summarize this information in a 2-state Markov chain Unt1 = Avn. Be sure
to identify the states of the system and to write down the transition matrix A,
and to explain what the entries of Vn mean.
2
(b) Assume that each province has 100,000 people today. How many people are
expected be in each province 2 years from now?
5
(c) Verify that A is power convergent and find limno+ An. Assuming that the
above trend continues, what is the population distribution of the country in the
long run?
5. Consider the matrix A 2
[4]
(a) Find the eigenvalues of A and calculate their algebraic and geometric multi-
plicities.
[2]
(b) Determine the Jordan canonical form J of A.
[4]
(c) Find an invertible matrix P such that P-1AP - J.
[10]
6. Find the characteristic polynomial and all the generalized geometric multiplicities
VA(12) of the matrix
-1
(GID)
and use these to determine the Jordan Canonical Form J of A.
Hint: Use the fact that
-CMC
and that A3=0.]
Page 3 of 4 pages
[10] 7. The matrix
3
2
21
3
1
2
3
has characteristic polynomial [You do not need to verify
this fact. Show that A is power convergent, and calculate
lim A".
n-00
[2]
8.
(a) Define what is meant by the orthogonal projection of a vector Rn onto a
subspace VCR".
[5]
(b) Let A and B be two mxm matrices which are similar. Prove that An is
similar to Bn, for all n 1, and that chA(t) = chB(t).
[3]
(c) State the Cayley-Hamilton Theorem and verify it for the matrix
^-(-12)
A

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