Math 307: Problems for section 1.3
1. Write down the vector approximating f (x) at interior points, the vector approximating
xf (x) at interior points, and the nite dierence matrix equation for the nite dierence
approximation with N = 4 for the dierential
Math 307: Problems for section 4.14.2
March 17, 2009
1. For the following matrices nd
(a) all eigenvalues
(b) linearly independent eigenvectors for each eigenvalue
(c) the algebraic and geometric multiplicity for each eigenvalue
and state whether the matr
Math 307: Problems for section 4.2
November 14, 2012
1. (i) What can you say about the diagonal elements of a Hermition matrix?
(ii) Show that if A is an n n matrix such that v, Aw = Av, w then A is Hermitian.
(i) Diagonal entries of Hermitian matrices ar
Math 307: Problems for section 4.2
1. (i) What can you say about the diagonal elements of a Hermition matrix?
(ii) Show that if A is an n n matrix such that v, Aw = Av, w then A is Hermitian.
(i) Diagonal entries of Hermitian matrices are real, because fo
Math 307: Problems for section 2.1
October 16, 2014
0
0
1
1
1
2 0 1 0 4
1. Are the vectors 1, 2, 3 , 2, 9 linearly independent? You may use MAT
2 1 2 0 7
3
1
0
1
1
LAB/Octave to perform calculations, but explain your answer.
Put the vectors in the
Math 307: Problems for section 4.2
November 23, 2009
1. The matrix
6
1
A = 2
2
1
1
5
2
1
2
2
2
3
1
2
2
1
1
3
2
1
2
2
2
3
has positive eigenvalues. Use the power method to nd the largest and the smallest
ones, and the corresponding eigenvectors. Check whet
Math 307: Problems for section 2.2
October 16, 2012
Problem: The following formula matrix occurs in a chemical system given by a rock sample
[3]. The elements are Si, Al, Fe, Mg, K, H and O. The species are
qu = quartz (SiO2 )
si = sillimanite (Al2 SiO5 )
Math 307: Problems for section 2.1
October 4, 2009
1
1
1
0
0
2 0 1 0 4
1. Are the vectors 1, 2, 3 , 2, 9 linearly independent? You may use MAT
2 1 2 0 7
1
1
0
1
3
LAB/Octave to perform calculations, but explain your answer.
Put the vectors in the c
Math 307: Problems for section 1.1
1. Use Gaussian elimination
1 2 3
1 2 3
(a) A =
5 6 7
5 6 7
to nd the solution(s) to Ax = b where
1 1
1
1
1
4
1 1 1 1
4
b = 1 ,
(b) A =
1 1 0
1
0
8
0 0
1
1
1
8
3
1
b = .
0
1
The process of Gaussian elimination i
Math 307: Problems for section 1.2
Many problems in this homework make use of a few MATLAB/Octave .m les that are provided on
the website. In order to use them, make sure that the les are in the same directory that you are running
MATLAB/Octave from (to s
Math 307: Problems for section 1.2
February 2, 2009
Many problems in this homework make use of a few MATLAB/Octave .m les that are provided on
the website. In order to use them, make sure that the les are in the same directory that you are running
MATLAB/
Math 307: Problems for section 1.3
1. Write down the vector approximating f (x) at interior points, the vector approximating
xf (x) at interior points, and the nite dierence matrix equation for the nite dierence
approximation with N = 4 for the dierential
Math 307: Problems for section 2.3
October 16, 2012
1. Let D be the incidence matrix in the example done in the course notes.
1 1
0
0
0 1 1
0
0 1 1
D= 0
0 1 0
1
1
0
0 1
Using MATLAB/Octave (or otherwise) compute rref (D) and nd the bases for N (D),
R(D
Math 307: Assignment 6
Please look at the Course News to see which of the follwing problems need to be solved. Denitely,
solving all of the problems would be helpful for the midterm exam but it is not mandatory.
1. Compute the 1,2 and innity norms for the
Math 307: Assignment 4
1. Let D be the following incidence matrix:
1 1
0
0
0 1 1
0
0 1 1
D= 0
0 1 0
1
1
0
0 1
Using MATLAB/Octave (or otherwise) compute rref (D) and nd the bases for N (D),
C (D) and C (DT ). Find a basis for N (DT ) by computing rref
Math 307: Problems for section 3.2
March 7, 2011
1. Review of complex numbers:
(a) Show that |zw| = |z|w| for any complex numbers z and w.
(b) Show that zw = z w for any complex numbers z and w.
(c) Show that z z = |z|2 for every complex number z.
(a) If
Math 307: Problems for section 2.1
October 4, 2009
0
1
0
1
1
2 0 1 0 4
1. Are the vectors 1, 2, 3 , 2, 9 linearly independent? You may use MAT
2 1 2 0 7
3
1
1
0
1
LAB/Octave to perform calculations, but explain your answer.
2. Which of the followin
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Math 307: Problems for section 3.13.2
March 3, 2009
1. Use the CauchySchwarz inequality for real vectors to show
2
x+y
( x + y )
2
Under what circumstances is the inequality an equality?
x + y, x + y = x, x + x, y + y, x + y, y (linearity of the inner pro
Math 307: Problems for section 3.33.5
March 16, 2009
1. Review of complex numbers:
(a) Show that |zw| = |z |w| for any complex numbers z and w.
(b) Show that zw = z w for any complex numbers z and w.
(c) Show that z z = |z |2 for every complex number z .
The Big Picture
We want to solve the boundary value problem
f (x) = r(x),
on the interval [0, 1] with boundary conditions f (0) = A and f (1) = B, where r(x) is a known
function. To solve the dierential equation means to nd f (x).
If we can solve the BDV
2
3
0
0
6x 3
2
plot(XL,YL)
hold off
I.2.6. The linear equations for cubic splines: version 2
This version follows Numerical Recipes in C, by Press, et. al.
Given points (x1 , y1 ), . . . , (xn , yn ) with x1 < x2 < < xn we wish to nd a collection of cubic
4.3: 11, 12, 14
11. Show that if A and B are similar matrices then det(A) = det(B).
Proof. Suppose A and B are similar matrices. Then B = S 1 AS for some
nonsingular matrix S. Since S is nonsingular, we know that det(S) 6= 0 and
1
det(S 1 ) = det(S) . Thu
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October 26, 2016 Math 307 Name: . V Page 2 out of 11
Problem 1. [21] Short Answer. The questions (a)-(g) below are worth 3 marks each and require
a short explanation (unless otherwise stated).
(a) Specify the size and value of the variable z that you wo
Math 307: Problems for section 1.2
Many problems in this homework make use of a few MATLAB/Octave .m files that are provided on
the website. In order to use them, make sure that the files are in the same directory that you are running
MATLAB/Octave from (
Math 307: Problems for section 3.3
March 15, 2011
1. Show that for v, w Cn
kv + wk2 = kvk2 + kwk2 + 2 Re(hv, wi)
sh is
ar stu
ed d
vi y re
aC s
o
ou urc
rs e
eH w
er as
o.
co
m
and use this to prove the polarization identity
hv, wi =
1
kv + wk2 kv wk2 + i
Math 307: Problems for section 1.1
1. Use Gaussian elimination
1 2 3
1 2 3
(a) A =
5 6 7
5 6 7
to find the solution(s) to Ax = b where
1 1
1
1
1
4
1 1 1 1
4
b = 1 ,
(b) A =
1 1 0
1
0
8
0 0
1
1
1
8
3
1
b=
0 .
1
sh is
ar stu
ed d
vi y re
aC s
o
ou
Math 307: Problems for section 4.14.2
March 17, 2009
1. For the following matrices find
(a) all eigenvalues
(b) linearly independent eigenvectors for each eigenvalue
sh is
ar stu
ed d
vi y re
aC s
o
ou urc
rs e
eH w
er as
o.
co
m
(c) the algebraic and geo