Math H54 Final Exam
December 14, 2011
Professor Michael VanValkenburgh
Name:
Student ID:
Instructions: Show all of your work, and clearly indicate your answers. Use the backs of pages as scratch
paper. You will need pencils/pens and erasers, nothing more.
1. Determinant is the product of eigenvalues. Let A be an n n matrix, and let (A) be its
characteristic polynomial, and let 1 , . . . , n be the roots of (A) counted with multiplicity. Show that
det(A) = 1 2 n
i.e. the determinant is the product of the ei
MATLAB code for plotting convergence of Fourier series
% Step function
N=100;
x=(-pi+(0:N-1)*(2*pi)/N)';
fx=zeros(size(x);
fx(find(x<0)=-1;
fx(find(x>=0)=1;
M=3*N;
xext=(-3*pi+(0:M-1)*(6*pi)/M)';
fext=[fx; fx; fx];
for l = 1 : 100
figure(1)
clf
hold on
pl
Lin Lin, Evans 1083
Email: linlin@math.berkeley.edu
http:/math.berkeley.edu/~linlin/H54
Math H54: Sample Midterm II, Fall 2014
This is a closed book, closed notes exam. You need to justify every one
of your answers unless you are asked not to do so. Compl
Lin Lin, Evans 1083
Email: linlin@math.berkeley.edu
http:/math.berkeley.edu/~linlin/H54
Math H54: Sample Midterm I, Fall 2014
This is a closed book, closed notes exam. You need to justify every one
of your answers unless you are asked not to do so. Comple
Lin Lin, Evans 1083
Email: linlin@math.berkeley.edu
http:/math.berkeley.edu/~linlin/H54
Math H54: Solution to Sample Midterm
II, Fall 2014
This is a closed book, closed notes exam. You need to justify every one
of your answers unless you are asked not to
Lin Lin, Evans 1083
Email: linlin@math.berkeley.edu
http:/math.berkeley.edu/~linlin/H54
Math H54: Solution to Sample Midterm
II, Fall 2014
This is a closed book, closed notes exam. You need to justify every one
of your answers unless you are asked not to
Lin Lin, Evans 1083
Email: linlin@math.berkeley.edu
http:/math.berkeley.edu/~linlin/H54
Math H54: Sample Final, Fall 2014
This is a closed book, closed notes exam. You need to justify every one
of your answers unless you are asked not to do so. Completely
Let V be the vector space consisting of all two by two matrices, and W be the vector space consisting
of one by one matrices. Dene
1
T (X) = 1 2 X
1
1.
2.
3.
4.
5.
Find a basis for V and W , and compute dim(V ) and dim(W ).
Show that T is a linear transfo
Math H54 Midterm 1
September 20, 2010
Professor Michael VanValkenburgh
Name:
Student ID:
Instructions: Show all of your work, and clearly indicate your answers. Use the backs of pages as scratch
paper. You will need pencils/pens and erasers, nothing more.
Math H54 Midterm 1
September 20, 2011
Professor Michael VanValkenburgh
Name:
Student ID:
Instructions: Show all of your work, and clearly indicate your answers. Use the backs of pages as scratch
paper. You will need pencils/pens and erasers, nothing more.
MATH H54, HOMEWORK 1
Exercise 1. According to wikipedia1, a penny is 2.5 grams and a nickel is 5 grams.
(a) Find a general expression for the number p of pennies and the number n of nickels
needed to have c cents and m grams.
(b) How many pennies and nick
MATH H54, HOMEWORK 2, DUE FRIDAY, SEPTEMBER 16, 2011
Exercise 1. We dene a (binary) relation on a set S to be a function R from the set S S
of ordered pairs to the set cfw_0, 1:
R:
S S cfw_0, 1.
When R(x, y ) = 1, we write xRy . A relation R is called an
MATH H54, HOMEWORK 3, SELECTED SOLUTIONS/HINTS
Exercise 1. Let
1
2 10
A = 1 0 3 5 .
1 2 1 1
Find a row-reduced echelon matrix R which is row-equivalent to A and an invertible
matrix P such that R = P A. Then nd P 1 .
Answer. There are at least two ways to
MATH H54, HOMEWORK 6, DUE MONDAY, OCTOBER 17, 2011
Exercise 1. Suppose A is a 22 matrix with real entries that is symmetric (that is, AT = A).
Prove that A is similar to a diagonal matrix.
Answer. By considering the characteristic polynomial, we nd that t
MATH H54, HOMEWORK 7, DUE MONDAY, OCTOBER 24, 2011
Exercise 1. Prove the following polarization identity : for all x, y Rn ,
1
1
x y = |x + y |2 |x y |2 .
4
4
Answer. We write it out in terms of dot products:
|x + y |2 |x y |2 = |x|2 + 2x y + |y |2 |x|2 +
MATH H54, HOMEWORK 8, DUE MONDAY, OCTOBER 31, 2011
Exercise 1. A scientist obtains the m data points
(t1 , y1 ), . . . , (tm , ym ) R2 .
The method of least squares gives a line of best t y = ct + d to this data set.
(a) Show that the normal equation AT A
MATH H54, HOMEWORK 9, DUE MONDAY, NOVEMBER 14, 2011
Exercise 1. Consider the ordinary dierential equation (ODE)
()
u + pu + qu = 0,
where p and q are constants. Suppose that the auxiliary equation has a repeated root
r. Then
y (t) = c1 ert + c2 tert
is a
MATH H54, HOMEWORK 10, DUE WEDNESDAY, NOVEMBER 23, 2011
Exercise 1. Consider the following scalar dierential equation:
x 2x + x = 0.
(a) Convert the scalar equation into an equivalent system, and, for any given initial
x
point
, nd the solution in the for
Lin Lin, Evans 1083
Email: linlin@math.berkeley.edu
http:/math.berkeley.edu/~linlin/H54
Math H54: Final Exam, Fall 2014
This is a closed book, closed notes exam. You need to justify every one
of your answers unless you are asked not to do so. Completely c