Teaching Assistant

M427J Differential Equations with Linear Algebra

Syllabus: See Homepage and Calendar.

• Regular Class: MWF 10:00-11:00 am @ JGB 2.216
• Discussion Session: TTh 3:30-4:30 pm @ JGB 2.216
• Dr. Meth’s Office Hour: MWF 11:05-11:55 @ RLM 9.144
• My Office Hour: W 12:00-2:00 pm @ RLM 11.130

Download sample solution to Braun Section 3.1, Question 4, 6 here.

Some Notations

Logic and Set Theory

• $\Rightarrow$ means implies. (Statement P) $\Rightarrow$ (Statement Q) means from statement P we can deduce Q, or we can say if P then Q.

• $\in$ means belongs to or inside. For an object $x$ and a set $A$, $x \in A$ means $x$ is an element of set $A$.

• $\subset$ and $\subseteq$ means subset. For two sets $A$ and $B$, $B \subset A$ or $B \subseteq A$ means $B$ is a subset of $A$, that is, every element of $B$ is an element of $A$, or we can write $x \in B \Rightarrow x \in A$.

• $\subsetneqq$ means proper subset. For two sets $A$ and $B$, $B \subsetneqq A$ means $B$ is a subset of $A$, but $B \neq A$.

• $\forall$ means for all. $\forall x \in A \dots$ means for all elements $x$ in the set $A$ we have $\dots$.

• $\exists$ means there exists. $\exists x \in A \dots$ means there exists an element $x$ in the set $A$ such that $\dots$.

• $f: A \to B$ means $f$ is a function with domain $A$ and target $B$.

Number Sets

• $\mathbb Z$ is the set of all integers.

• $\mathbb N$ is the set of all natural numbers (including 0).

• $\mathbb Q$ is the set of all rational numbers.

• $\mathbb R$ is the set of all real numbers.

• $\mathbb C$ is the set of all complex numbers (including real numbers).

Linear Algebra

• $\mathbb R ^n$ is the set of all $n$ dimensional real vectors.

• $\mathbb C ^n$ is the set of all $n$ dimensional complex vectors.

• $\mathcal{M} _{r \times c}$ is the set of all $r$ by $c$ matrices.

• $\mathcal{M} _{n}$ is the set of all $n$ by $n$ matrices.

• For a matrix $A$, $\mathrm{Ker} (A)$ or $\mathrm{Nul} (A)$ is the Kernel of $A$, that is the solutions to $A\boldsymbol{x}=\boldsymbol{0}$.

Function spaces

• $C ^0$, $C (\mathbb R)$, $C ^0 (\mathbb R)$, $C (\mathbb R, \mathbb R)$, $C ^0 (\mathbb R, \mathbb R)$ is the set of continuous functions from $\mathbb R$ to $\mathbb R$, that is, continuous real-valued functions.

• $C ^k$, $C ^k (\mathbb R)$, $C ^k (\mathbb R, \mathbb R)$ is the set of $k$ time continuously differentiable functions, that is, continuous functions whose first through $n$th derivatives are continuous.

• $C ^\infty$, $C ^\infty (\mathbb R)$, $C ^\infty (\mathbb R, \mathbb R)$ is the set of smooth functions, that is, any order of derivative is continuous.

• $C (\mathbb R, \mathbb R ^n)$, $C ^0 (\mathbb R, \mathbb R ^n)$ is the set of continuous functions from $\mathbb R$ to $\mathbb R ^n$, that is, continuous vector-valued functions.

• $C ^k (\mathbb R, \mathbb R ^n)$ is the set of $k$ time continuously differentiable vector-valued functions.

• $C ^\infty (\mathbb R, \mathbb R ^n)$ is the set of smooth vector-valued functions, that is, any order of derivative is continuous.

• $\mathcal{P} _m$, $\mathcal{P} _m[t]$ is the set of all polynomials of $t$ with real coefficients and degree less or equal than $m$.

• $\mathcal{P}$, $\mathcal{P}[t]$ is the set of all polynomials of $t$.

Differential Calculus

If $y(t)$ is a differentiable function of $t$, we write

$$\dot y = y' = \frac{\mathrm{d}y}{\mathrm{d}t}$$

to represent derivative of $y$ with respect to $t$.

• If $L$ is a linear differential operator, $V _L$ is the set of solutions to $L[y] = 0$.

• If $L=\frac{\mathrm d}{\mathrm dt}-A$, so the equation is $\dot{\boldsymbol{x}} = A \boldsymbol{x}$, $V _A$ is the set of solutions to $\dot{\boldsymbol{x}} = A \boldsymbol{x}$.