Mathematics Department Stanford University
Math 51H Homework 9 Solutions
1. Suppose A, B are similar nn matrices (i.e. there is an nn invertible matrix C with B = C 1 AC).
Prove that A, B have the same eigenvalues.
Hint: Show that det(B I) det(A I) R.
Sol
Mathematics Department Stanford University
Math 51H Homework 2
Solutions
1. Use Q.8(b) of hw1 to prove the cosine law, that if A, B, C are distinct points in Rn and if is
the angle (as dened in lecture) between the vectors AB and AC then BC 2 = AB 2 + AC
Mathematics Department Stanford University
Math 51H Vector spaces and linear maps
We start with the denition of a vector space; you can nd this in Section A.8 of the text (over R,
but it works over any eld).
Denition 1 A vector space (V, +, ) over a eld F
Here are two equivalent denitions for a k-dimensional C 1 submanifold of Rn .
(From the book.) A subset M Rn is a k-dimensional C 1 submanifold of Rn i for each
point x M there is a positive real number > 0 such that the intersection B (x) M is
a permuta
Mathematics Department Stanford University
Math 51H Homework 8
Solutions
1. (i) Prove that the degree n polynomials u, v obtained by taking the real and imaginary parts of
2
2
(x + iy)n are harmonic (i.e. xp + yp 0 on R2 in both cases p = u, p = v).
2
2
H
Mathematics Department Stanford University
Math 51H Homework 7
Solutions
1. (i) Use 1-variable calculus to prove that if f (x) is dierentiable for 0 < x < a and continuous for
0 x a, and if f (x) 0 for all 0 < x < a, then f (x) is increasing on [0, a] (i.
Mathematics Department Stanford University
Math 51H Homework 6
Solutions
1. Use the chain rule and any other theorems from lecture that you need to show the following:
(i) If f : U V and g : V Rp are C 1 , where U Rn , V Rm are open, then g f is C 1 on U
Mathematics Department Stanford University
Math 51H Homework 5
Solutions
1. Recall that for a given subset A Rn , we dened E A to be relatively open in A if for each
x E there is > 0 such that B (x) A E.
Prove that E A is relatively open in A an open set
Mathematics Department Stanford University
Math 51H Homework 3
Solutions
1. Weve seen in lecture that if A is an m n matrix then matrix multiplication Ax (where x Rn )
is a linear operation (i.e. A(x + y) = Ax + Ay for all x, y Rn and all , R).
(a) Prove
Mathematics Department Stanford University
Math 51H Homework 1 Solutions
1. Explain why the following proof of the Cauchy-Schwarz inequality |x y| x
x, y Rn , is not valid:
y , where
Proof: If either x or y is zero, then the inequality |x y| x y is trivia
Mathematics Department Stanford University
Math 51H Homework 9
Due at TA Section Friday Nov. 30
1. Suppose A, B are similar nn matrices (i.e. there is an nn invertible matrix C with B = C 1 AC).
Prove that A, B have the same eigenvalues.
Hint: Show that d
Mathematics Department Stanford University
Math 51H Homework 10
Not to be graded; solutions will be posted Wednesday Dec. 5
1. (i) If U Rn is open, f : U Rn is C 1 , and det Df (x) = 0 for each x U , prove that f (U ) is open.
(ii) In case f is as in (i)
How to:
Find the nullspace of a matrix
Put A in row reduced echelon form, with pivot columns j1 , . . . , jr . Then row i expresses the
pivot variable xji as a linear combination of the non-pivot variables. Together, we combine this
expressions into a ve
Mathematics Department Stanford University
Math 51H Homework 10
Solutions
1. (i) If U Rn is open, f : U Rn is C 1 , and det Df (x) = 0 for each x U , prove that f (U ) is
open. (ii) In case f is as in (i) and satises the additional hypothesis that it is a
Mathematics Department Stanford University
Math 51H Homework 4
Solutions.
1. Let [0, 2) and let T be the linear transformation of R2 dened by T (x) = Q()x, where Q()
cos sin
r cos
is the 2 2 matrix
. Prove that if x =
(with r 0 and [0, 2) then
sin
cos
Mathematics Department Stanford University
Math 51H Contraction mapping theorem and ODEs
The contraction mapping theorem concerns maps f : X X, (X, d) a metric space, and their xed
points. A point x is a xed point of f if f (x) = x, i.e. f xes x. A contra
Mathematics Department Stanford University
Math 51H Distributions
Distributions rst arose in solving partial dierential equations by duality arguments; a later related
benet was that one could always dierentiate them arbitrarily many times. They also prov
Mathematics Department Stanford University
Math 51H Integrals
There is a class of functions on which one knows what the integral should be. Namely, if f : [a, b] R
is an ane function, i.e. f (t) = t + for some , R, then one should have
b
f=
a
a+b
+ (b a)
Mathematics Department Stanford University
Math 51H Duals, adjoints/transposes and the spectral theorem
Suppose V is an n-dimensional vector space over a eld F such as R. When discussing determinants,
we have shown that the vector space of m-linear maps,
Mathematics Department Stanford University
Math 51H Basic algebra
We start with the denition of a group, since it involves only one operation.
Denition 1 A group (G, ) is a set G together with a map : G G G with the properties
1. (Associativity) For all x
Mathematics Department Stanford University
Math 51H Mean value theorem, Taylors theorem and integrals
If f is a C 1 real valued function on an open set U Rn , we have for any x, h, i, with h suciently
small, that
f (x + hei ) f (x) = hf (x + hei )
for som
Mathematics Department Stanford University
Math 51H Chain rule
As a warm up to the chain rule, lets talk about the composition of continuous functions.
Theorem 1 Suppose (X, dX ), (Y, dY ), (Z, dZ ) are metric spaces, f : X Y is continuous at a X,
g : Y Z
Mathematics Department Stanford University
Math 51H Inner products
Recall the denition of an inner product space; see Appendix A.8 of the textbook.
Denition 1 An inner product space V is a vector space over R with a map , : V V R such
that
1. (Positive de
Mathematics Department Stanford University
Math 51H Metric spaces
We have talked about the notion of convergence in R:
Denition 1 A sequence cfw_an of reals converges to R if for all > 0 there exists N N
n=1
such that n N, n N implies |an | < . One write
Mathematics Department Stanford University
Math 51H Open and closed sets
We saw that Bolzano-Weierstrass in R implies the compactness of intervals [a, b]. The analogue of
Bolzano-Weierstrass in Rn also holds:
Theorem 1 Suppose cfw_x(k) is a bounded seque
Mathematics Department Stanford University
Math 51H Homework 8
Due at TA section, Friday Nov. 16
*Reminder: Mid-term 2 is Tuesday Nov. 13, 7:158:30pm*
1. (i) Prove that the degree n polynomials u, v obtained by taking the real and imaginary parts of
2
2
(
Mathematics Department Stanford University
Math 51H Homework 6
Due at TA section, Friday Nov. 2
1. Use the chain rule and any other theorems from lecture that you need to show the following:
(i) If f : U V and g : V Rp are C 1 , where U Rn , V Rm are open
Mathematics Department Stanford University
Math 51H Homework 7
Due at TA section, Friday Nov. 9
1. (i) Use 1-variable calculus to prove that if f (x) is dierentiable for 0 < x < a and continuous for
0 x a, and if f (x) 0 for all 0 < x < a, then f (x) is i
Mathematics Department Stanford University
Math 51H Homework 4
Due at TA section, Friday Oct. 18
Reminder: Mid-term 1, Tue Oct 15, 7:15pm
1. (a) Let [0, 2) and let T be the linear transformation of R2 dened by T (x) = Q()x, where
cos sin
r cos
Q() is th
Mathematics Department Stanford University
Math 51H Mid-Term 1
October 15, 2013
Unless otherwise indicated, you can use results
covered in lecture and homework, provided they are clearly stated.
If necessary, continue solutions on backs of pages
Note: wor