Lecture 01

[[lecture-data]]

2024-08-26

Readings

0. Chapter 0

Field

A field is a set on which addition, subtraction (additive inverse), multiplication, and division (multiplicative inverse) exist and behave "nicely"

Example

$R$ or $C$

(see field)

Vector Space

We can add vectors and scale vectors by scalars and they are still in the space. And we have a zero vector :)

Example

$R^{n}$ , $C^{n}$ , $C^{'} [a, b]$
For arbitrary field, we denote this $F$

(see vector space)

Matrix notation

In this class we have $M_{m, n} (R)$ (textbook) is a real $m \times n$ matrix. No parentheses indicates complex, and only one subscript indicates a square matrix

Linear Combination

Let $V$ be a vector space over a field $F$ . We say that the vector consisting of any sum of the vectors $v_{1}, v_{2}, \dots, v_{k}$ with scalings $α_{1}, α_{2}, \dots, α_{k}$ is a linear combination of the vectors $v_{i}$ .

α_{1} v_{1} + α_{2} v_{2} + \dots + α_{k} v_{k}

(see linear combination)

Span

The span ${v_{1}, v_{2}, \dots, v_{k}} := {β_{1} v_{1} + β_{2} v_{2} + \dots + β_{k} v_{k}; β_{i} \in F}$

That is, all linear combinations of the vectors $v_{i}$

(see span)

Note

Note that the zero vector is always in the span of some set of vectors

Linearly Dependent

$v_{1}, v_{2}, \dots, v_{k}$ are linearly dependent means that $\exists γ_{1}, γ_{2}, \dots, γ_{k} \in F$ (not all zero) such that $γ_{1} v_{1} + γ_{2} v_{2} + \dots + γ_{k} v_{k} = 0$

(They are linearly independent if there exist no such $γ_{1}, γ_{2}, \dots, γ_{k}$ ; ie that $γ_{1} v_{1} + γ_{2} v_{2} + \dots + γ_{k} v_{k} = 0 ⟹ γ_{1} = \dots = γ_{k} = 0$ )

(see also linear dependence)

Theorem

$v_{1}, v_{2}, \dots, v_{k} \in V$ are linearly dependent if and only if one of the vectors $v_{i}$ is a linear combination of the others.

(see linear dependence)

Basis

$v_{1}, v_{2}, \dots, v_{k}$ are a basis for $V$ means two things:

$v_{1}, v_{2}, \dots, v_{k}$ are linearly independent
$Span {v_{1}, v_{2}, \dots, v_{k}} = V$

(see basis)

Note

If $v_{1}, \dots, v_{k}$ is a basis for $V$ , then for every vector $w \in V$ there exist UNIQUE scalars $δ_{1}, δ_{2}, \dots, δ_{k} \in F$ such that $$\delta_{1}v_{1} + \dots + \delta_{k}v_{k} = w$$

(see bases create unique definitions for vectors in their space)

Intuition (uniqueness)
Suppose we have $ϵ v_{1} + ϵ v_{2} + \dots + ϵ v_{k} = w$ and also $δ_{1} v_{1} + \dots + δ_{k} v_{k} = w$ . Then we have that $0 = w - w = (ϵ_{1} - δ_{1}) v_{1} + \dots + (ϵ_{k} - δ_{k}) v_{k} = 0$
But since $v_{1}, \dots, v_{k}$ is a basis, this implies that we must have $ϵ_{i} - δ_{i} = 0 \forall i$

Note that the number of vectors in the vector space depends only on the vector space $V$ itself (this is called "dimension", lets call it $k$ ). This means that $V$ is isomorphic with the field $F^{k}$ . That is, that there exists a bijection between $V$ and $F^{k}$ (but this requires axiom of choice)

There is a dual nature to matrices $M_{m, n} (F)$

Algebraic structure ("vector-like")
Analytic structure ("function-like")
We associate any matrix $A \in M_{m, n} (F) \leftrightarrow A^{'} : F^{n} \to F^{m}$
And we say $\forall x \in F^{n}$ we have $A^{'} (x) := A x$

Such a function is linear.

Linear Function

A function $T : V \to W$ linear means that $\forall x, y \in V, α, β \in F$ , we have

T (α x + β y) = α T (x) + β T (y)

(see linear function)

Linear Function

Suppose our field is $C^{'} [a, b]$ (differentiable functions on the interval $[a, b]$ ). What is an example of a linear operator here?

The derivative!

[α f (t) + β g (t)]^{'} = α f^{'} (t) + β g^{'} (t)

Suppose I have two vector spaces $V$ and $W$ with finite dimensions $n$ and $m$ respectively. Now suppose we have a linear function $T : V \to W$ . We can think of $V$ as column vectors with $n$ coordinates and $W$ as column vectors with $m$ coordinates.

We can come up with a matrix that describes the function by taking the standard basis for $V$ and taking its mapping in $W$ . We can then use the mappings as the columns of the matrix.

If we take any basis vector, we will get the correct output by construction
Then since it is a linear operator, we get any other output also by construction