HOMEWORK SET 4
DUE SUNDAY MARCH 08 VIA GRADESCOPE BY 11:59 PM
These questions stem from the text book, Differential Equations and Their Applications by Martin
Braun.
Your score on this and subsequent homework assignments will be based on one randomly chosen
problem. You will be allowed to drop one homework as indicated in the syllabus.
Question 4.1. Systems of Linear Equations. Solve the following system of linear equations:
x + 2y − z = 1,
2x + 5y + z = 4,
−x + y + 2z = 0.
Write the solution in vector form.
Question 4.2. Matrix Form of a System. Write the following system in the form Ax = b and
identify the matrix A and vectors x and b:
3x − y + 2z = 5,
x + 4y − z = −1,
2x + y + z = 3.
Question 4.3. Vector Operations. Let
1
u = −2 ,
3
2
v = 1 .
−1
Compute:
(1) 2u − v
(2) u · v
Question 4.4. Vector Space Verification. Let
x
V = y ∈ R3 : x + y + z = 0 .
z
Show that V is a vector space by verifying closure under addition and scalar multiplication.
Question 4.5. Subspaces. Determine whether the following set is a subspace of R2 :
x
W =
: y = 2x + 1 .
y
Justify your answer.
1
Question 4.6. Linear Independence. Determine whether the vectors
1
2
−1
v1 = 0 , v2 = 1 , v3 = 1
2
4
−2
are linearly independent or linearly dependent.
Question 4.7. Linear Dependence. Find constants a and b such that
3
1
1
6 = a 2 + b 1 ,
9
3
1
or explain why this is not possible.
Question 4.8. Span of Vectors. Determine whether the vector
4
w = 1
3
lies in the span of
1
v1 = 0 ,
1
1
v2 = 1 .
1
Question 4.9. Eigenvalues and ODE Motivation. Consider the matrix
2 1
A=
.
1 2
Find the eigenvalues of A. Explain briefly why eigenvalues are important in solving systems of
linear differential equations
x′ = Ax.
Question 4.10. Homogeneous Systems. Consider the system of ODEs
1 2
′
x =
x.
2 4
Explain how linear dependence of the rows of the matrix affects the solution structure of the system.
2
0
