Page 322 - 35Linear Algebra

322 Sample First Midterm

(d) What are the vectors X 0 and Y i called and which matrix equations do
they solve?
(e) Check separately that X 0 and each Y i solve the matrix systems you
claimed they solved in part (d).

3. Use row operations to invert the matrix

 
1 2 3 4
2 4 7 11
 
 
3 7 14 25
 
4 11 25 50

2 1 T −1
4. Let M = . Calculate M M . Is M symmetric? What is the
3 −1
2
trace of the transpose of f(M), where f(x) = x − 1?
5. In this problem M is the matrix

cos θ sin θ
M =
− sin θ cos θ
and X is the vector

x
X = .
y
Calculate all possible dot products between the vectors X and MX. Com-
pute the lengths of X and MX. What is the angle between the vectors MX
and X. Draw a picture of these vectors in the plane. For what values of θ
do you expect equality in the triangle and Cauchy–Schwartz inequalities?

6. Let M be the matrix
1 0 0 1 0 0
 
0 1 0 0 1 0
 
 
0 0 1 0 0 1
 
 
 0 0 0 1 0 0 
 
 0 0 0 0 1 0 
0 0 0 0 0 1
k
Find a formula for M for any positive integer power k. Try some simple
examples like k = 2, 3 if confused.
7. What does it mean for a function to be linear? Check that integration is a
linear function from V to V , where V = {f : R → R | f is integrable} is a
vector space over R with usual addition and scalar multiplication.

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