Math 104, HW 3, Due 4/29 before class
Please write down your name and Stanford ID and staple your solutions. Solutions are
due before the class.
1. Prove that if x, y are two non-zero vectors in R2 ,
Math 104
Winter 2012
Homework 3: solution set
Matrices are bold capital, vectors are bold lowercase and scalars or entries are not bold.
Problem 1
Since x y C we have (x y ) = (x y ) and so y x = (x y
MATH 104: APPLIED MATRIX THEORY
HOMEWORK ASSIGNEMENT 9 - SOLUTIONS
Problem 1
In this exercise, we are interested in constructing a factorisation AQ = LU , where Gaussian elimination
is carried out wit
Math 104 Spring 2016 Homework 8 Solution
May 25, 2016
Question 1 For a quadratic polynomial y = c0 + c1 x + c2 x2 , the error of the estimate of y in terms of x
with this polynomial is given by kAc bk
Math 104 Spring 2016 Homework 2 Solution
April 28, 2016
Question 1 Suppose A is an m n matrix. By the algorithm described by the textbook, the singular
values of A can be found inductively by maximizi
Math 104, HW 1, Due 4/15 before class
Please write down your name and Stanford ID and staple your solutions. Solutions are
due before the class.
1. Prove that if C, v Cn , and v = 0, then either = 0 o
Math 104, HW 2, Due 4/22 before class
Please write down your name and Stanford ID and staple your solutions. Solutions are
due before the class.
1. Suppose that A Rpq is injective and (v1 , . . . , vn
Math 104, HW 5, Due 5/13 before class
Please write down your name and Stanford ID and staple your solutions. Solutions are due
before the class.
1. Consideer the matrix
3 2
A = 4 0
0 1
Compute a redu
Math 104, HW 4, Due 5/6 before class
Please write down your name and Stanford ID and staple your solutions. Solutions are
due before the class.
1. Suppose that U is a subspace of Rn . Prove that
dim U
Math 104, HW 6, Due 5/20 before class
Please write down your name and Stanford ID and staple your solutions. Solutions are due
before the class.
1. Let A R22 be the matirx
Determine A
11
Determine A
Math 104, HW 8, Due 6/3 before class
Please write down your name and Stanford ID and staple your solutions. Solutions are due
before the class.
1. A matrix A Cmm is normal if A A = AA . Show using the
Math 104, HW 7, Due 5/27 before class
Please write down your name and Stanford ID and staple your solutions. Solutions are due
before the class.
1. Suppose that A Rmn is full rank with m > n. Suppose
MATH104 Homework 2 Solutions
Yang Zhou
1. Suppose that for some scalars c1 , , cn we have
c1 Av1 + c2 Av2 + + cn Avn = 0
We want to prove that c1 = c2 = = cn = 0.
Using the linearity of A we can rewri
Math104 Spring 2014 Homework1 Solutions
Yang Zhou
Problem 1 Suppose v = (v1 , ., vn ). Then v = 0 means vk = 0 for all
k = 1, ., n. If = 0, nothing needs to be shown. If = 0, then vk = 0 implies
that
Math104 Spring 2014 Homework3 Solutions
Yang Zhou
April 30, 2014
Problem 1 If either x = 0 or y = 0, both sides are 0 and we are done. Now
lets suppose they are nonzero. Set u = x/ x and v = y/ y , we
MATH 104: APPLIED MATRIX THEORY
HOMEWORK ASSIGNEMENT 3 - SOLUTIONS
Problem 1 (T&B 3.1)
Let k k be a vector norm and W a nonsingular matrix in Cmm . For x Cm , we need to check that
kxkW = kW xk is a v
MATH 104: APPLIED MATRIX THEORY
HOMEWORK ASSIGNEMENT 7 - SOLUTIONS
Problem 1
1
1
2
Let us denote a1 = 2i , a2 = 2i and a3 = 0 .
2
2
4
We see that a3 = a1 + a2 . As we have only 2 linearly independan
Math 104
Winter 2012
Homework 4: solution set
Problem 2
Consider the 2 2 matrix
P=
1
0
1
.
0
This matrix obeys P 2 = P but is not orthogonal since it is not symmetric. This matrix projects points onto
Math 104
Winter 2012
Homework 5: solution set
Problem 1
(a) By computing the eigenvalues and eigenvectors of A A, we have the SVD A = U V , where
2
2
U=
22
22
2
2
10 2
, =
0
3
0
, V = 54
5
52
4
5.
Math 104
Winter 2012
Homework 6: solution set
Problem 1, 7.1
1
(a) Suppose that A = 0
1
0
1 = [a1 , a2 ]. We will use the Classical Gram-Schmidt method to calculate
0
2
2
a1
0
the reduced and full Q
Math 104
Winter 2012
Homework 7: solution set
Matrices are bold capital, vectors are bold lowercase and scalars or entries are not bold.
Problem 2
Since A is Hermitian, the spectral theorem asserts th
Math 104
Winter 2012
Homework 8: solution set
Problem 1
There are n n matrices D diagonal and V invertible such that A = V DV 1 . The elements on the
diagonal of D are the eigenvalues of A denoted by
1
Math 104
Agenda
Inexact computation
Bad pivoting
Rooting of polynomials
2
Nnumerical stability
Computer arithmetic is inexact (nite memory)
Issues arise from inexact computations
Interested in
Math 104 Fall 2008
Class notes
Laurent Demanet
Draft December 1, 2008
2
Preface
These class notes complement the class textbook Numerical Linear Algebra
by Trefethen and Bau. They do not suce in thems
Agenda
Math 104
1
Google PageRank algorithm
2
Developing a formula for ranking web pages
3
Interpretation
4
Computing the score of each page
Google: background
Mid nineties:
many search engines
often
Math 104
Winter 2012
Unitary matrices
As we have seen in class, an n n unitary matrix U is a matrix obeying
U U = I .
(1)
Expressed dierently, a unitary matrix is a matrix whose inverse is its adjoint
MATH 104: APPLIED MATRIX THEORY
HOMEWORK ASSIGNEMENT 5 - SOLUTIONS
Problem 1
The set cfw_y + N (A) is the set of all possible solutions to the linear system Ax = b.
To prove this, we show that any vec
Math 104 Spring 2016 Homework 6 Solution
May 11, 2016
Question 1 We first prove that for an n n matrix A, A has a left inverse if and only if rank(A) = n. Let
a1 , . . . , an be column vectors of A. I
Math 104 Spring 2016 Homework 2 Solution
April 13, 2016
Question 1 Without loss of generality we only prove y x = 0 when x y = 0. Suppose x y = 0. Take
transpose, we have 0 = y t (
xt )t = y t x
. Tak