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
. Take conjugate, get (
y )t x = 0, as desired.
t
Now suppos
MATH 104: APPLIED MATRIX THEORY
HOMEWORK ASSIGNEMENT 1 - SOLUTIONS
Problem 1 (T&B 1.1)
(a) Recall from MATH51 (or equivalent) that there exist standard matrices allowing to produce the
operations asked in this problem (multiplication by scalar, row/column
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 independant vectors in the list, the basis we are
looking for wil
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 vector norm. We verify the 3 properties of vector norms:
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 with pivoting by columns (instead of rows).
(a) Assume tha
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, where
50
1 1 1
30
1 2 4
c
0
A=
1 4 16 , c
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 maximizing kAxk under kxk = 1 and x orthogonal to existing
righ
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 or v = 0.
2. Prove that if (v1 , . . . , vn ) spans a ve
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 ) is linearly independent in Rq .
Prove that (Av1 , .
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 , then
x, y = x
y cos
where is the angle between x and y
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 reduced QR factorization of A
Solve the least square probl
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 = n dim U
2. Suppose that U is a subspace of Rn . Prov
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
1 2
0 1
and nd the vectors u with u
1
and nd the vecto
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. If LA = I, then Laj = ej , where ej is the j-th standard
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 vector of the form y + x0 where x0 N (A) is solution of th
Math 104
winter 2012
Homework 1: solution set
Problem 2
7
1
4
If 8 is in the span of 1 and 1 then there will exist and such that
9
3
2
7
1
4
1
5
8 = 1 + 1 = ( ) 1 + 0 .
9
3
2
3
5
The middle row give = 8 and replacing in the previous system
15
5
0 = 0 ,
3
Math 104
winter 2012
Homework 2: solution set
We use the notations below to ease readability. Matrices are bold capital, vectors are bold lowercase
and scalars or entries are not bold. For instance, A is a matrix and aij (sometimes, A(i, j ) its (i, j )th
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 ).
Applying this result give y Ax = y (Ax) = (Ax) y =
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
the horizontal line in a direction parallel to the 45
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.
3
5
Denote U = [u1 , u2 ] and V = [v1 , v2 ]. The SVD
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 QR decomposition of A. We begin with r11 = a1 = 2, q1 =
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 that A = U U , where U is a unitary matrix and
= diag(1
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 j . Then
det(A) = det(V ) det(D ) det(V 1 ) = det(V V 1
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 robust and stable algorithms
3
Number representation
F
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 themselves as study material;
it is only in combination with
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 times not that eective
Late nineties:
Google goes onlin
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. If the columns of U are the
vectors u1 , . . . , un ,
Math 104: Lecture 1
Agenda:
Course organization
Why linear algebra?
Course objectives
Matrix vector products
Least-squares problem
Course organization
http:/www-stat.stanford.edu/~candes/math104/index.shtml
Why linear algebra?
Fundamental tool to understa
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 Schur factorization that if
A is normal then A can be