Lecture Notes

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====== Homework 2 ======
{{ :math121a-f23:math121a_hw2.pdf | 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?
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