Table of Contents
Homework 2
solution (thanks to an anonymous student who provided the solution)
I will update the homework after each lectures. It is due next Wednesday (since we have Labor day Monday)
Vector Space Problems
1. Let $V \In \R^3$ be the points that $\{(x_1, x_2, x_3) \mid x_1 + x_2 + x_3=0\}$. Find a basis in $V$, and write the vector $(2,-1,-1)$ in that basis.
2. Let $V$ as above,. Let $W = \R^2$, let $V \to W$ be the map of forgetting coordinate $x_3$. Is this an isomorphism? What's the inverse?
3. Let $V$ as above, and let$W$ be the line generated by vector $(1,2,3)$. Let $f: V \to W$ be the orthogonal projection, sending $v$ to the closest point on $W$. Is this a linear map? How do you show it? What's the kernel? Let $g: W \to V$ be the orthogonal projection. Is it a linear map? What's the relationship between $f$ and $g$?
4. about quotient space. Let $V = \R^2$, and let $W$ be the linear subspace generated by vector $(1,2)$ (i.e. the line passing through origin and $(1,2)$). For $v \in V$, let $[v] = v+W\in V/W$ denote the equivalence class that $v$ belongs to, i.e., the (affine) line parallel to $W$ and passing through $v$. Draw some pictures to answer these questions.
- Is it true that $[(0,0)] = [(1,2)]$?
- Is it true that $[(1,1)] = [(0,1)]$?
5. another important notion is dual vector space. Given a vector space $V$, the dual vector space is $V^* = Hom(V, \R)$, the set of linear maps from $V$ to $\R$ (Hom is short for 'homomorphism', which means linear maps for vector spaces). For example, if $V = \R^2$, the linear functions $x$ and $y$ belong to $V^*$, we have $V^* = \{ax + by \mid a,b \in \R\}$. Here $x,y$ are basis for $V^*$.
Let $V$ be the vector space of polynomials with degree less or equal than 3. What's the dimension of $V$? What's the dimension of $V^*$? Can you find a basis for $V$? A basis for $V^*$?
Calculus
1. Here is claim $1+2+3+4+\cdots = -1/12$. Show that this is wrong.
Fun fact: There is an interesting function , called Riemann Zeta function $\zeta(s)$, which for $s > 1$ can be written as $\zeta(s) = \sum_{n=1}^\infty 1/n^s$. In fact $\zeta(s)$ is actually a meromorphic function of $s$, and $\zeta(-1) = -1/12$.
2. Does the following series converge? Explain why.
- $\sum_{n=1}^\infty 1/n^2$.
- $\sum_{n=1}^\infty 1/n!$
- $\sum_{n=1}^\infty n^2/n!$
3. Let $a_n$ be a sequence of $\pm 1$. Show that $\sum_{n=1}^\infty a_n / 2^n$ is convergent. (Hint: absolute convergence implies convergence)
4. What is radius of convergence? Is it true that $$ \frac{1}{1-x} = 1+ x + x^2 + \cdots $$ holds for all real number $x \neq 1$?
5. We know that the following series diverge $$ 1 + 1/2 + 1/3+ 1/4 \cdots. $$ Question: does the following alternating series converge? Why? $$ 1 - 1/2 + 1/3 - 1/4 + \cdots $$ (Optional): Fix any real number $a$. Show that by rearrange the order of the terms in the above alternating series, we can have the series converges to $a$.
6. Line integral: let $\gamma$ be the straightline from $(0,0)$ to $(1,1)$. Compute the line integral $$ \int_\gamma 2 dx + 3 dy. $$ What if we replace $\gamma$ by a curved line but still from $(0,0)$ to $(1,1)$, would the above result change? Why?
September 1: Differentiation and Integration
Differentiation
What is differentiation? it is measuring the ratio of how the output change versus how the input changes. It is a linear map from the vector space of small change of input, to the vector space of small changes of output.
Chain Rule
There is also the chain rules, which says, if quantity x affect y,and y affect z, then x affects z. If $y=2x$, $z=3y$, then $z = 6x$.
Partial Derivative
If you have a function $f(x,y)$ that depends on two input variables, you can ask how sensitive the output is on each of them, say $$ \frac{\d f}{\d x}(x_0, y_0) = \lim_{\epsilon \to 0} \frac{f(x_0 + \epsilon, y_0) - f(x_0, y_0)}{\epsilon} $$
Integration
What is integration? It is a process of collecting stuff / contributions along the way.For example, the integral (when $f(x)$ is continuous) is the limit of the following approximations $$ \int_a^b f(x) dx = \lim_{N\to \infty} f(x_{N,i}) \Delta_N x, \quad x_{N,i} = a + \frac{b-a}{N} i, \Delta_N x = \frac{b-a}{N}. $$
The fundamental theorem of calculus says $$ \int_a^b f'(x) dx = f(b) - f(a).$$
- line integral
- Green's theorem.
August 30: Review of Calculus
today we will go over sequence of numbers and limit.
sequence
Let $a_1, a_2, \cdots $ be a sequence of numbers. We can have many examples of it.
- $1,-1,1,-1\cdots $
- $0.9,0.99, 0.999, $
- $1,2,3, \cdots, $
limit
We say a sequence $(a_n)$ converges to $a$, if for any $\epsilon>0$, there exists $N>0$, such that for any $n > N$, we have $|a_n - a| \leq \epsilon$.
series
a series is something that looks like $\sum_{n=1}^\infty a_n$. We can define the partial sum $S_n = \sum_{j=1}^n a_j$. We say the series $\sum_n a_n$ convergers if and only the partial sum converges.
Series is like a discretized version of integral.
various tests for series convergence
1. what does absolute convergence mean for series?
2. the model convergent series
- $\sum_n 1/n^p$ for $p > 1$.
- $\sum_n 1/r^n$ for $r>1$
3. various tests
- comparison test, if $0<a_n < b_n$, and $\sum_n b_n$ converges, then $\sum_n a_n$ converges.
- ratio test
- root test
exercises
some exercise from Boas's textbook, try 1,3, 4-8
August 28: Review Linear algebra 2
Today we will continue our review of linear algebra. Hopefully you have brushed up on the set notations over the weekends.
quotient space
Let $V$ be a vector space, and $W \In V$ be a subspace. The quotient space $V/W$ is the following vector space:
- as a set, it consist of element that looks like the shifted linear subspaces $[v] = v+W =\{v + w \mid w \in W\}$. We note that, if $v_1, v_2 \in V$ satisfies $v_1 - v_2 \in W$, then $[v_1]=[v_2]$.
- the vector space operations are defined using the representatives, namely
- $ c [v]= [cv], \forall c \in \R, v \in V$
- $[v_1] + [v_2] = [v_1 + v_2]$
Motivation: why we care about quotient space? What's the meaning of the subspace? When we quotient something out, we are defining some equivalence relation, and we are ignoring some differences. In the quotient vector space case, suppose we want to identify the vectors in space $W$ as $0$, we say two points $v_1, v_2$ are equivalent, if their difference is in $W$.
basis
A basis in $V$ is a collection of vectors, such that they are maximally linearly independent.
Given a basis, we can express all other vectors using linear combination of the basis. The coefficiients in the linear combination are called coordinates.
inner product
An inner product on a vector space is a function $(,): V \times V \to \R$, such that
- symmetric: $(v,w) = (w,v)$
- linear in each slot: $(av_1 + bv_2, w) = a (v_1, w) + b(v_2, w)$ (by symmetry, also linear in the 2nd slot)
- positive definite. $(v,v) \geq 0$ and $=0$ only if $v=0$.
You may be very familiar with the notion of $\R^n$, equipped with a (default) Euclidean inner product. But in general, for a vector space $V$, the inner product is something that you give it afterwards.
A nice basis for vector space with inner product is called an orthonormal basis. $e_1, \cdots, e_n \in V$, such that $(e_i,e_j) = \delta_{ij}$.
Orthogonal projection
If $V$ is a vector space with an inner product, and $W \In V$ is a subspace, then we can define some projection $$\pi: V \to W $$ it satisfies that $ v - \pi(v) \perp \pi(v). $
Exercise time
Let $V \In \R^3$ be the points that $\{(x_1, x_2, x_3) \mid x_1 + x_2 + x_3=0\}$. Find a basis in $V$, and write the vector $(2,-1,-1)$ in that basis.
Let $W = \R^2$, let $V \to W$ be the map of forgetting coordinate $x_3$. Is this an isomorphism? What's the inverse?
Let $V$ as above, and $W$ be the line generated by vector $(1,2,3)$. Let $f: V \to W$ be the orthogonal projection, sending $v$ to the closest point on $W$. Is this a linear map? How do you show it? What's the kernel? Let $g: W \to V$ be the orthogonal projection. Is it a linear map? What's the relatino between $f$ and $g$?
Review of Linear Algebra
Let's start from scratch again. What is linear algebra?
This is a textbook on linear algebra by Prof Givental.
What is a vector space
Answer 1
row vectors, column vectors, matrices. Let's also review the index notation $a_i = \sum_{j} M_{ij} b_j$
Very concrete, very computable.
Answer 2
Geometrical, as we went over in class.
A 2-dimentional vector is something you can draw.
A 3-dim vector, hmm, harder.
how about 4-dim vector? $\infty$-dim one? It doesn't matter the dimension, the rule we obtain from 2 and 3 dimensional one is good enough.
Answer 3
the goofy math prof: a vector space is a set $V$ together with two operations
- scalar multiplication: given a number $c$ and a vector $v \in V$, we need to specify the output $c v \in V$.
- vector addition: given two vectors $v_1, v_2 \in V$, we need to specify the output $v_1 + v_2 \in V$
such that, some obvious conditions should be satisfied.
Why we care about this? Because it is somehow useful.
For example,
- the subspace of $\R^3$ that is perpendicular to $(1,2,3)$ forms a vector space.
- other examples? non-examples?
Linear Map
How does vector space talk to each other? Linear map.
Do an example of stretching, skewing.
Do a non-example of bending a line in $\R^2$.
Kernel, image and cokernel
We didn't quite cover the idea of a quotient space. I will do that next time.
