MATH 200 - LINEAR ALGEBRA HOMEWORK 6
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MATH 200 - LINEAR ALGEBRA
HOMEWORK 6, DUE Monday, December 11, 2023, at 11:59pm
Upload your written (or typed) solutions for this assignment to the appropriate place in the Spring by the due date. You may work on this assignment with your classmates, but you must write up your own solutions.
Upload your Mathematica input and output to the Spring as well, either as a pdf file or an nb file.
Part A. (10 points)
Problem 1. Let A = (a1 , a2 , a3 ) and B = (b1 , b2 , b3 ) be two ordered bases for a vector sapce V.
Suppose
a1 = 3b1 − b2 ,
a2 = −b1 + b2 + b3 , and
a3 = b2 − 4b3 .
(a) Find the change-of-coordinates matrix from A to B.
(b) Find [x]B for x = −2a1 + a2 − 3a3 .
Problem 2. Let ⃗ube a vector in R2 , and let ⃗v be another vector in R2 which is perpendicular to
⃗u. Let L be the line through the origin in the direction of ⃗u, and let T : R2 → R2 be reflection across the line L.
(a) Find all eigenvaluesof T.
(b) For each eigenvalue of T, find a corresponding eigenvector.
Problem 3. Let
A = l1(0) 0 35(0 0)
.
0 1 2 l
Find a diagonal matrix D and an invertible matrix P such that A = PDP−1 .
Part B. (10 points)
In this problem we describe the power method for numerically approximating eigenvalues of a matrix. This method is simple but very useful, and is used within many important algorithms, including Google’s PageRank algorithm.
Definitions. For this algorithm we will need two definitions. The Mathematica implementation of these definitions is also described below.
1. Let
⃗x = 
x(x)2(1)
[
nl
be a vector in Rn. The norm of ⃗x is
I⃗xI = 4x1(2) + x2(2) + ··· + xn(2) ,
and the normalization of ⃗x is ⃗x/I⃗xI. For example, the norm of ⃗x = [ 21] is " [ 21] " =^22 + ( — 1)2 =^5
,
so the normalization of ⃗x is
= 
21] = [ 21(/)/^(^5)5] .
To compute the normalization of a vector x in Mathematica, we use the command Normalize[]. For example, to find the normalization of ⃗x = [ 21], we use
Normalize[{2.0,-1.0}].
2. Let
⃗x = 
x(x)2(1)
and ⃗y = 
y(y)2(1)
[
nl [
nl
be two vectors in Rn. The dot product of ⃗x and ⃗y, denoted ⃗x · ⃗y, is
⃗x · ⃗y = x1y1 + x2y2 + ··· + xn yn.
This can be computed in Mathematica using the . command. For example, to find the dot
product of ⃗x = [2(1)] and ⃗y = [ 32], we use the command
{1,2}.{3,-2}.
The Power Method. Let A be an n X n matrix. The power method is used to approximate the dominant eigenvalue for A, that is, the eigenvalue with the greatest absolute value. The method goes as follows:
1. Pick a random vector ⃗x0 in Rn.
2. Iteratively define ⃗xi to be the normalization of A⃗xi-1 for all i > 0.
As i gets large, ⃗xi converges to aneigenvector of A corresponding to the largest eigenvalue of A. To approximate the eigenvalue itself, we use the Rayleigh quotient. If ⃗xi isan approximation of aneigenvector for A, then an approximation of the corresponding eigenvalue is found by computing
A⃗xi · ⃗xi .
⃗xi · ⃗xi
An Example. Let us give an example of the first three iterations of the power method. Let
A = [0(2) 1(1)] .
We want to approximate the largest eigenvalue for A. Let us start with the vector
⃗x0 = [ 1(1)] .
Then, following the power method described above, we define
⃗x1 = ∥A(A)
0(0) ∥ = [0(0).(.)316228(948683)] .
Continuing:
⃗x2 = ∥A(A)
1(1) ∥ = [0(0).(.)141421(989949)] ,
and
⃗x3 = ∥A(A)
2(2) ∥ = [0(0).(.)066519(997785)] .
(In fact, we can see that these vectors seem to be converging to the vector [0(1)], which is an
eigenvector for A.)
To find the corresponding eigenvalue, we compute the Rayleigh quotient:
= 2.06195.
This suggests that 2 is probably an eigenvalue for A, and indeed this is the case.
oC(b)o(l)mput(em 1)e(.) 
⃗(x) 
. 
5]. Let ⃗x0 = [ 1(1)], and define ⃗xi as in the power method above.
(b) Use the Rayleigh quotient for ⃗x10 to approximate the largest eigenvalue of A.
Problem 2. Let B = l 5(4) 3(0) 2(0) 
. Let ⃗x0 = l 1(1)
, and define ⃗xi as in the power method above.
−2 0 −2
1
(b) Use the Rayleigh quotient for ⃗x10 to approximate the largest eigenvalue of A.
2023-11-29