Download Differential Equations and Linear Algebra Test #2 Review

Differential Equations and Linear Algebra
Test #2 Review
Some problems are from Elementary Differential Equations, Boyce and DiPrima and Linear
Algebra and its Applications, Lay.
1. (Systems of Equations) Find the critical points and nullclines for the system of equations.
Are they stable or unstable?
x0 = x(3 − 2x − y)
y 0 = y(8 − 4y − 3x)
2. (Matrix operations) Calculate the following quantities, or state

4
1 2
8 4 6
C = 2
B=
A=
3 4
4 2 5
1
a)AB
e) det(B)
b) BA
f)rref(A)
c) AC
g) C −1
that it is impossible and why.

6
7
4
d)det(A)
h) B −1
3. (Matrix Operations, Solution Sets) Solve the following systems. If there are multiple
solutions, describe the solution set in the form x = xp + xh .


1 5 2 −6 9
0
0 0 1 −7 4 −8 

a) 
2 10 5 −19 22 −7  x = 0
1 5 4 −20 17 −16


 
1 4 2
12
b) 2 14 10 x = 50
2 10 9
41


 
2 2 4
4



c) 1 3 2 x = 8
2 3 4
7


 
1 3 0 −3
2



3 x = 11
d) 2 6 2
1 3 −2 1
0
1
4. (Matrix inverse, Matrix operations) Find the

1
1
1
inverse and determinant of:

0 1
2 1
4 2
5. (Cramer’s Rule) Solve the system using Cramer’s Rule. For what values of k is there a
unique solution?
5x1 + 7x2 = 3
2x1 + kx2 = 1
6. (Vector Spaces) Are the following vector spaces? Prove if they are, and provide a
counterexample if not.
(a) W = {functions f such that f = ax2 for a ∈ R}
(b) W = {f ∈ P3 such that p(0) = 0}


−2t
(c) Vectors of the form  0  such that t ∈ R
t
(d) Polynomials of degree 3 with integer coefficients.
 
t
(e) Vectors of the form 5 such that t ∈ R
t


a−1
(f) Vectors of the form a − 6b such that a, b ∈ R
2b + a


2−a
(g) Vectors of the form  4b  such that a, b, c ∈ R
b + 3c
2
7. (Linear Independence, span and basis) Are the following a basis for the indicated space? If
not, are they linear dependent or so they fail to span the space?
(a) {1, x2 + x, x + 1}. V = P3
     
0
0 
 1





0 , 0 , 1 , V = R3
(b)


1
0
0
(c) {4x, 2x2 + x, x, 3x3 + 1}, V = P3
     
1
1 
 1
(d) 0 , 2 , 1 , V = R3


0
0
1
8. True or False?
(a) A single vector is linearly dependent.
(b) The columns for an nxn invertible matrix form a basis for Rn .
(c) If a finite subset S spans a vector space V , then some subset of S is a basis for V .
(d) R2 is a subspace of R3 .
(e) W is a subspace of V if it is closed under scalar multiplication.
(f) If two rows of a 3x3 matrix A are the same, then det(A)=0.
(g) If A, B are nxn matrices, then det(A + B) = det(A)+det(B).
(h) If det(A) = 3, then det(A3 ) = 27.
(i) If A3 = 0, then det(A) = 0.
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