Lecture 1 - Syllabus and Whatnot

Syllabus

Jeffery has office hours (see Canvas). Wednesdays are pretty quiet so if you work on any specific day that's the best time to hang out. If there's any cool places to study let him know as those Wednesday office hours may change location due to construction.

How the Course is Structured?

Each week has a HW set, and each set is due the following week Thursday. Hence, you have until next Thursday to do the next HW. But, please start it early as a result, since there's really no late submissions or anything like that. The time is there to pause and ponder, not to procrastinate.

NOTE: no in-class meeting on Thursday. Liese will be out of town.

We also have Quizzes on Thursday. On weeks we don't have quizzes we have MT exams (also Thursday). There also might be a strike, which would reschedule everything by about a week.

The First Few Sections

You start always at the axioms for a proofs based course. That's what we'll do today. First, let's look at the objects from MATH 206 (Linear Algebra I):

• Vectors
• Matrices
• Determinants
• Linear Independents
• Bases
• ...

Recall we'd try to solve $A\overrightarrow{x}=\overrightarrow{b}$. Each $\overrightarrow{x},\overrightarrow{b}$ are vectors. $A$ transforms $\overrightarrow{x}$ into $\overrightarrow{b}$. $A$ (when it's a matrix) allows for a linear transformation. We're going to study vector spaces, and the linear transformations on those spaces. This is the study of linear algebra (by definition).

What is a Vector?

Sometimes people just say a vector is a quantity that has a magnitude and a direction. We can:

• Add and subtract vectors: $\overrightarrow{x}+\overrightarrow{y}$
• Scalar Multiply Vector by $\lambda$: $\lambda \overrightarrow{c}$
• Notions of products:
  • Dot Product: $\overrightarrow{a}\cdot \overrightarrow{b}$
  • Cross Product: $\overrightarrow{a}\times \overrightarrow{b}$.
• We can normalize vectors and define orthogonality (perpendicularity).

Recall ${\mathbb{R}}^n=\{(x_1,...,x_n{)}^T|x_1,...,x_n\in \mathbb{R}\}$. We know that ${\mathbb{R}}^n$ is a vector space. Mathematicians have decided these definitions because in problems of use, these are the things they needed. They had use cases for these, hence where the definition comes from. Really, mathematicians said that you only need:

• The addition of vectors together
• The scalar multiplication by some scalar $\lambda$.

For these vectors, we require a certain list of properties in order for the vector space name to hold. Mathematicians gathered the similarities and found the following properties:

![[VectorSpaceDefinition.pdf]]
When $\mathbb{F}$ is the real, then $V$ is a real-vector space. And if $\mathbb{F}=\mathbb{C}$ then it's a complex vector space.