Lecture Notes

$$\gdef\E{\mathbb E}$$ Due Date : May 10th (Sunday) 11:59PM. Submit online to gradescope.

Policy : You can use textbook and your notes. There should be no discussion or collaborations, since this is suppose to be a exam on your own understanding. If you found some question that is unclear, please let me know via email.

All vectors spaces are finite dimensional over $\R$.

1. True or False (10 pts)

1. Any vector space has a unique basis.
2. Any vector space has a unique inner product.
3. Given a basis $e_1, \cdots, e_n$ of $V$, there exists a basis $E^1, \cdots, E^n$ on $V^*$, such that $E^i (e_j) = \delta_{ij}$.
4. If we change $e_1$ in the basis $e_1, \cdots, e_n$, in the dual basis only $E^1$ will change.
5. Given a vector space with inner product, there exists a unique orthogonal basis.
6. Let $V$ and $W$ be two vector spaces with inner products and of the same dimension.Then there exists a linear map $f: V \to W$, such that for any $v_1, v_2 \in V$, $$\la v_1, v_2 \ra = \la f(v_1), f(v_2) \ra.$$
7. If $V$ and $W$ are vector spaces of dimension $3$ and $5$, then the tensor product $V \otimes W$ have dimension $8$.
8. If $V$ has dimension $5$, then the exterior power $\wedge^3 V$ is a vector space with dimension $10$.
9. The solution space of equation $y'(x) + x^2 y(x) = 0$ forms a vector space.
10. The solution space of equation $y'(x) + x y^2(x) = 0$ forms a vector space.

2. (10 pt) Let $V_n$ be the vector space of polynomials whose degree is at most $n$. Let $f(x)$ be any smooth function on $[-1,1]$. We fix $f(x)$ once and for all. Show that there is a unique element $f_n \in V_n$ (depending on our choice of $f$), such that for any $g \in V_n$, we have $$ \int_{-1}^1 f_n(x) g(x) dx = \int_{-1}^1 f(x) g(x) dx. $$

Hint:

• (1) Equip $V_n$ with an inner product $\la g_1, g_2 \ra = \int_{-1}^1 g_1(x) g_2(x) dx$.
• (2) Show that $f(x)$ induces an element in $V_n^*$: $g \mapsto \int_{-1}^1 f(x) g(x) dx$.
• (3) use inner product to identify $V_n$ and $V_n^*$.

A remark: if $f(x)$ were a polynomial of degree less than $n$, then you could just take $f_n(x) = f(x)$. But, we have limited our choices of $f_n$ to be just degree $ \leq n$ polynomial, so we are looking for a 'best approximation' of $f(x)$ in $V_n$ in a sense. Try solve the example case of $n=1$, $f(x) = \sin(x)$ if you need some intuition.

3. (10 pt) Let $\R^3$ be equipped with curvilinear coordinate $(u,v,w)$ where $$ u = x, v = y, w = z - x^2 + y^2. $$

1. (3pt) Write the vector fields $\d_u, \d_v, \d_w$ in terms of $\d_x, \d_y, \d_z$.
2. (3pt) Write the 1-forms (co-vector fields) $du,dv,dw$ in terms of $dx, dy, dz$.
3. (4pt) Write down the standard metric of $\R^3$ in coordinates $(u,v,w)$.

1. (10 pt) Orthogonal polynomials. Let $I = [-1,1]$ be a closed interval. $w(x) = x^2$ a non-negative function on $I$. For functions $f,g$ on $I$, we define their inner products as $$ \la f, g \ra = \int_{-1}^1 f(x) g(x) w(x) dx $$ The normalized orthogonal polynomials $P_0, P_1, \cdots$ are defined by

1. $P_n(x)$ is a degree $n$ polynomial.
2. $\la P_n, P_n \ra = 1$
3. $\la P_i, P_j \ra = 0$ if $i \neq j$.

Find out $P_0, P_1, P_2$.

2. (10 pt) Find eigenvalues and eigenfunctions for the Laplacian on the unit sphere $S^2$, i.e., solve $$ \Delta F(\theta, \varphi) = \lambda F(\theta, \varphi) $$ for appropriate $\lambda$ and $F$. The Laplacian on a sphere is $$ \Delta f = \frac{1}{\sin \theta} \d_\theta(\sin \theta \d_\theta(f)) + \frac{1}{\sin^2 \theta} \d_\varphi^2 f. $$

3. (5 pt) Find eigenvalues and eigenfunctions for the Laplacian on the half unit sphere $S^2$ with Dirichelet boundary condition, i.e., solve $$ \Delta F(\theta, \varphi) = \lambda F(\theta, \varphi), \quad F(\theta=\pi/2, \varphi)=0. $$ for appropriate $\lambda$ and $F$.

4. (15 pt) (Heat flow). Consider heat flow on the closed interval $[0,1]$ $$ \d_t u(x,t) = \d_x^2 u(x,t), $$ where $u(x,t)$ denote the temperature. \ Let $u(0, t) = u(1, t) = 0$ for all $t$. Let the initial condition be $$ u(x, 0) = \begin{cases} 2x & x \in [0, 1/2] \cr 2(1-x) & x \in [1/2, 1] \end{cases} $$

• (12pt) Solve the equation for $t > 0$.
• (3 pt) Does the solution make sense for any negative $t$? Why or why not?

5. (10 pt) (Steady Heat equation). Let $D$ be the unit disk. We consider the steady state heat equation on $D$ $$ \Delta u(r, \theta) = 0 $$

• (3 pt) Write down the Laplacian $\Delta$ in polar coordinate
• (5 pt) Show that, if the boundary value is $u(r=1, \theta) = 0$, then $u=0$ on the entire disk.
• (2 pt) Is it possible to have a boundary condition $u(r=1, \theta) = f(\theta)$, such that there are two different solutions $u_1(r,\theta)$ and $u_2(r,\theta)$ to the problem?

1. (5 pt) Throw a die 100 times. Let $X$ be the random variable that denote the number of times that $4$ appears. What distribution does $X$ follow? What is its mean and variance?

2. (5 pt) Let $X \sim N(0,1)$ be a standard normal R.V . Compute its moment generating function $$ \E(e^{t X}). $$ Use the moment generating function to find out $\E(X^4)$. Let $Y = X^2$. What is the mean and variance of $Y$?

3. (5 pt) There are two bags of balls. Bag A contains 4 black balls and 6 white balls, Bag B contains 10 black balls and 10 white balls. Suppose we randomly pick a bag (with equal probability) and randomly pick a ball. Given that the ball is white, what is the probability that we picked bag A?

4. (5 pt) Consider a random walk on the real line: at $t=0$, one start at $x=0$. Let $S_n$ denote the position at $t=n$, then $S_n = S_{n-1} + X_n$, where $X_n = \pm 1$ with equal probability.

• (3pt) What is the variance of $S_n$?
• (2pt) Use Markov inequality, prove that for any $c > 1$, we have

$$ \P(|S_n| > c \sqrt{n}) \leq 1/c^2 $$