Homework 6 Solutions - ALL ANSWERS ARE CORRECT - University of California, Los Angeles LIFESCIENC 30B
1 view 0 purchase
Course
LIFESCIENC 30B mathmatics for life science
Institution
University Of California, Los Angeles
Homework 6 Solutions
Total points: 32
6.4.7: Show that M3 =
1 0
0 1
.
Solution:
M3 =
0 1
−1 −1
0 1
−1 −1
0 1
−1 −1
=
0 1
−1 −1
−1 −1
1 0
=
1 0
0 1
.
6.4.8: Using the point p =
5
0
as the test point, apply M three ti...
,6.4 FE 7: Compute the eigenvalues of the linear function
X 4X − 5Y
f( )=
Y 2X − 2Y
Solution: The matrix representation of this linear function is
4 −5
.
2 −2
The characteristic equation is
λ2 − 2λ + 2 = 0.
Therefore,
λ = 1 ± i.
−9 −8
6.4 FE 8: One of the eigenvalues of the matrix is 3. What is a corresponding eigen-
12 11
vector for it?
X −9 −8
Solution: Let’s denote by an arbitrary eigenvector with eigenvalue 3 of , so that
Y 12 11
−9 −8 X X
=3· .
12 11 Y Y
This gives you a system of two linear equations in X and Y , and you can solve the first or the
second equation to find that
3
Y = − X.
2
So, the vectors of the form
X
,
− 32 X
where X 6= 0, are the eigenvectors with eigenvalue 3 of this matrix. Letting X = 2, you can choose
as your particular eigenvector
2
V= .
−3
2
,
4 5 −3
6.4 FE 9: One of the eigenvalues of the matrix 4 6 −4, is 2. What is a corresponding
8 11 −7
eigenvector for it?
Solution: Let’s denote by
X
Y
Z
an arbitrary eigenvector with eigenvalue 2 of this matrix, so that
4 5 −3 X X
4 6 −4 Y = 2 · Y .
8 11 −7 Z Z
This gives you a system of three linear equations in X, Y, and Z, namely
4X + 5Y − 3Z = 2X
4X + 6Y − 4Z = 2Y
8X + 11Y − 7Z = 2Z
After simplification, you will find that
X = 2Y, Z = 3Y
(to get started, look at the coefficients of X). So, the vectors of the form
2Y
Y ,
3Y
where Y 6= 0, are the eigenvectors with eigenvalue 2 of this matrix. Letting Y = 1, you can choose
as your eigenvector
2
V = 1 .
3
6.4 FE 10:
a) Solve for a and b in the equation
2 −3 9
a +b =
5 1 14
9
b) Use your answer to part (a) to give the coordinates of with respect to the basis
14
2 −3
, .
5 1
3
The benefits of buying summaries with Stuvia:
Guaranteed quality through customer reviews
Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.
Quick and easy check-out
You can quickly pay through credit card or Stuvia-credit for the summaries. There is no membership needed.
Focus on what matters
Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!
Frequently asked questions
What do I get when I buy this document?
You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.
Satisfaction guarantee: how does it work?
Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.
Who am I buying these notes from?
Stuvia is a marketplace, so you are not buying this document from us, but from seller ExamsConnoisseur. Stuvia facilitates payment to the seller.
Will I be stuck with a subscription?
No, you only buy these notes for $9.99. You're not tied to anything after your purchase.
