Math 285
Spring 2014
Name ______________________
Final Exam
No Notes! No Cell Phone!
You can use the calculator to find REF, RREF. Do not use the calculator to integrate for
you. Do not use the calculator to find your eigenvalues – must see work!
1. Let V = {( a1, a2 ) : a1, a2 ∈ R, a2 > 0} . Define addition and scalar multiplication on
V as follows:
( a1, a2 ) + (b1, b2 ) = ( a1 + b1, a2 b2 )
k ( a1, a2 ) = ( ka1, a2k )
k∈R
{
}
Show that W = ( a, 2 a ) : a ∈ R is a subspace of V.
1
2. Use diagonalization method to solve
x1! = 6x1 − x2
x!2 = −5x1 + 2x2
2
3. Solve the differential equation (Hint: this is Bernoulli)
−1
−1
2
3
y! + 6x y = 3x y cos x,
x>0
3
4. Determine two linearly independent power series solutions to the given
differential equation centered at x = 0.
y!! + 2x 2 y! + 2xy = 0
4
5. Use Laplace Transforms to solve
y!! − 4y = 2tet
y (0) = 0
y! ( 0 ) = 0
5
6. Use variation of parameters method to solve
y!! −10 y! + 25y =
2e 5x
4 + x2
6
7. Let
T : P2 → M 2 ( R)
# −a − b 0 &
T ( a + bx + cx 2 ) = %
(
$ 3c − a −2b '
a) Determine if T is a linear transformation.
b) If it is a linear transformation, find a basis and hence the dimension of the
kernel and the range.
c) Is T 1-1, onto, both or neither?
7
8. Determine the general solution to
y!! + 2 y! + y = 3cos x + 4sin x
8
9. Decide (with justification) whether the given set S of vectors (a) spans V and (b)
is linearly independent.
V = P4
S = { x 4 + x 2 +1, x 2 + x +1, x +1, x 4 + 2x + 3}
9
10. A tank contains 8 L of water in which is dissolved 32 g of chemical. A solution
containing 2 g/L of the chemical flows into the tank at a rate of 4 L/min, and the
well-stirred mixture flows out at a rate of 2 L/min. Determine the amount of
chemical in the tank after 20 minutes.
10
11. Find all the eigenvalues and eigenvectors of
12 − 6⎤
⎡ 5
⎢
A = ⎢− 3 − 10 6 ⎥⎥
⎢⎣− 3 − 12 8 ⎥⎦
11
  
12. Let ( v1, v2 , v3 ) be three linearly independent vectors in a vector space V. Is the set
     
{v1 − v2 , v2 − v3, v3 − v1 } linearly dependent or linearly independent? Show some
work.
12
13. Solve the given system:
x1! = x1 + 2x2 + 5e 4t
x!2 = 2x1 + x2
13
14. Find the inverse Laplace transform for
F ( s) =
2s +1
2
( s −1) ( s + 2)
14