The bible stays the same: (Axler 1997)

We will be less exploratory, Axler will pretty much tell us. However, we should try to say stuff in the class every single class period.

There is a ban on numbers over 4 on this class.

Best Practices

• Ask questions
• Talk to each other
• Make mistakes
• From Riley: know the Proof Design Patterns

Non-Axler but Important

Things we explicitly are told to know, but is not immediately in Axler. You bet you determinants are going to be here.

A Linear Combination of vectors is a… guess what? Any vector formed by a combination of vectors at arbitrary scales.

constituents

• A list of vectors \(v_1, \dots,v_{m}\)
• Scalars \(a_1, \dots, v_{m} \in \mathbb{F}\)

requirements

A Linear Combination is defined formally by:

\begin{equation} v = a_1v_1+\dots+a_{m}v_{m} \end{equation}

Here it is:

\begin{equation} a\frac{dy}{dx} + by = c \end{equation}

For some constants \(a,b,c\). The name is pretty obvious, because we have constants and the highest power on everything is \(1\). Its first-order because the derivative is only the first-order derivative.

linear (diffeq)

We technically call it “linear” because: if there are two possible solutions \(y_1(x)\) \(y_2(x)\), a linear combination \(Ay_1(x)+By_2(x)\) should also be a solution. Its “linear” because linear combinations work.

Linear Dependence Lemma is AFAIK one of the more important results of elementary linear algebra.

statement

Suppose \(v_1, \dots v_{m}\) is an linearly dependent list in \(V\); then \(\exists j \in \{1, 2, \dots m\}\) such that…

1. \(v_{j} \in span(v_1, \dots, v_{j-1})\)
2. the span of the list constructed by removing \(v_{j}\) from \(v_1, \dots v_{m}\) equals the span of \(v_1, \dots v_{m}\) itself

intuition: “in a linearly dependent list of vectors, one of the vectors is in the span of the previous ones, and we can throw it out without changing the span.”

A linear map to numbers. Its very powerful because any linear functional can be represented as an inner product using Riesz Representation Theorem

constituents

• vector space \(V\)
• a linear map \(\varphi \in \mathcal{L}(V, \mathbb{F})\)

requirements

\(\varphi\) is called a linear functional on \(V\) if \(\varphi: V \to \mathbb{F}\). That is, it maps elements of \(V\) to scalars. For instance, every inner product is a Linear Map to scalars and hence a linear functional.

additional information

Riesz Representation Theorem

Suppose \(V\) is finite-dimensional, and \(\varphi\) is a linear functional on \(V\); then, there exists an unique \(u \in V\) such that: