MATH 3210 Class Notes 01-22-2025

notelinear-algebra

Abstraction

We will take concepts like ${\mathbb{R}}^n$ and matrices from MATH 2210 and abstract them to abstract vector spaces and linear maps.

Review: ${\mathbb{R}}^n$ and ${\mathbb{C}}^n$

$$\mathbb{C}=\{a+bi:i^2=\sqrt{-1},a,b\in \mathbb{R}\}$$

We will generally prefer working with ${\mathbb{C}}^n$ because $\mathbb{C}$ is algebraically closed.

We will note a generic field as $\mathbb{F}$, which we can usually think of as $\mathbb{R}$ or $\mathbb{C}$ (and sometimes ${\mathbb{F}}_{q^n}$, a field of prime order). Generic elements of $\mathbb{F}$ are scalars.

Properties of Complex Arithmetic

1. Commutativity $\alpha +\beta =\beta +\alpha \forall \alpha ,\beta \in \mathbb{C}$ $\alpha \beta =\beta \alpha$
2. Associativity $\alpha +(\beta +\gamma )=(\alpha +\beta )+\gamma \forall \alpha ,\beta ,\gamma \in \mathbb{C}$ $\alpha (\beta \gamma )=(\alpha \beta )\gamma$
3. Identities $\alpha +0=\alpha \forall \alpha \in \mathbb{C}$ $1\alpha =\alpha$
4. Inverses For all $\alpha \in \mathbb{C}$, $\exists -\alpha \in \mathbb{C}$ such that $\alpha +(-\alpha )=0$. For all $\alpha \ne 0\in \mathbb{C}$, $\exists \frac{1}{\alpha }\in \mathbb{C}$ such that $\alpha (\frac{1}{\alpha })=1$.
5. Distribution $\alpha (\beta +\gamma )=\alpha \beta +\alpha \gamma \forall \alpha ,\beta ,\gamma \in \mathbb{C}$.

Background

Definition: Let $n$ be a positive integer. ${\mathbb{F}}^n$ is the set of all $n$-tuples of elements of $\mathbb{F}$:

$${\mathbb{F}}^n=\{(x_1,x_2,\dots ,x_n):x_k\in \mathbb{F}\text{ for }K=1,\dots ,n\}.$$

We say $x_k$ is the $k$-th coordinate of $(x_1,\dots ,x_n)$.

Definition: Addition in ${\mathbb{F}}^n$ is done component-wise.

Proposition:

If $x,y\in {\mathbb{F}}^n$, then $x+y=y+x$.

Proof:

$$\begin{array}{rl}x+y& =(x_1,\dots ,x_n)+(y_1,\dots ,y_n),\\ & =(x_1+y_1,\dots ,x_n+y_n),\\ & =(y_1+x_1,\dots ,y_n+x_n),\\ & =(y_1,\dots ,y_n)+(x_1,\dots ,x_n).\end{array}$$