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Su10 Math 270
FINAL
Name: _____________________
Show all necessary work clearly, neatly, systematically and understandably. Any incorrect and/or
understatement may be penalized. Failure to use correct notation may result in point deduction. Provide
clear flow of thought in the presentation of proofing.
1 2 3
1. (9:5,4) Let A  2 5 3 .
1 0 8


a. Decompose A into LU.
2.
Let A be an m x n matrix. Prove:
a. The ATA and AAT are symmetric.
(9:4,5)
b. If ATA = A, then A is symmetric and A = A2.
b. Find A-1
3.
(6)
Prove: det[adj(A)] = [det(A)]n-1
Let L: R3  R3 be a linear operator that first
rotates a vector counterclockwise about the zaxis through an angle of 30° and then reflects
the resulting vector about the yz-plane. Find
the matrix A representing L.
4.
(6)
6.
(8)
Prove Fundamental Subspaces Theorem:
Let A be an m x n matrix. Then N(A)  C (AT ) 
5.
(6)
Prove: (A-1+B-1)-1 = A(A+B)-1B.
Hint: AA-1 = A-1A = I = BB-1 = B-1B
7.
(10)
Let S be a subspace of C[a,b],
S = span{ex,xex,x2ex}. Let D be a differentiation
operator of S. Find the matrix representing D2
with respect to [ex,xex,x2ex].
Note that D2 represents the second
differentiation operator of S.
8.
Let plane P have the equation x – 3y + 2z = 6 and point Q(2,-3,1). Find the point on plane P that is
closest to Q.
(7)
1
1
0




9. (8) Let b1   1  , b2   0  and b3   1  and L: R3  R2. L x1 e1  x 2 e2  x 3 e3  x 2  x1 e1  x 3  x 2 e2 .
0
1
1
 
 
 
 1 
1
IF u1    , u2    , then find matrix representing L with respect to b1 , b2 , b3 to u1 ,u2
  2
2



 

10. (8:3,5) The stock market performance on one day is either positive or negative. According to the statistics:
If it is positive on the previous day, then it is positive again today at 80% probability.
If it is negative on the previous day, then it is negative again today at 70% probability.
a. Construct the transition matrix.
b. In the long run, find the probability that it
will be positive.
11. (18:4,6,4,4) Let L: P3  P3 where Lp(x)  p' (x)  p(3) .
a. Show that L is a linear transformation.
b. Find the matrix A representing L with respect to [x2,x,1].
c. Find Ker(L)
d. Find L(P2). Note: P2 is a subspace of P3.
y 1 '  3y 1  y 2  2y 3

12. (12) Solve  y 2 '  2y 1  2y 3
for initial value y0  (1,6,5)T
 y 3 '  2y 1  y 2  y 3