Vector Spaces
- Basis
- Dimension
- Fundamental Matrix Subspaces
- Linear Dependence and Independence
- Polynomials as Vectors
- Spanning Sets
- Subspace
- Vector Spaces Over Complex Numbers
- Vector Spaces
For all intents and purposes, any theorem described with vector space
Abstract Vector Spaces
Apples are vectors
- Molino
What is a vector? Well it's not an apple. An arrow and a line? Something with direction, and magnitude? Turns out, you have to meet a certain set or criteria to actually be a vector. It's a pretty exclusive cult actually, kinda uncool.
We care, because when anyone makes some theorem or lemma, they need to be able use a certain set of agreed upon properties, so that the theorem works and can be applied to anything that's a vector.
So, a vector not an arrow and a line. It's just... anything that satisfies the following properties:
A set
| Property | English |
|---|---|
| addition is commutative | |
| addition is associative | |
| zero vector | |
| additive inverse | |
| scalar multiplication if associative | |
| distributive law | |
| distributive law | |
| scalar multiplicative identity |
Then the elements of
Usually, "vector space" will only refer to real numbers, but technically, complex numbers are included too.
Well, since we have no idea what's real and what's not anymore, let's see what is vector.
- The set
of all functions is a vector space - The set
of all continuous functions is a vector space - The set
of all differentiability functions is a vector space - In fact, differentiation is a linear transformation, and its inverse it the integral (well, not really, since
exists)
- In fact, differentiation is a linear transformation, and its inverse it the integral (well, not really, since
Try some of the properties on polynomials. For example,
Imagine that each coefficient of the polynomial is a vector component.
Consider the piecewise functions:
Both are discontinuous, but their sum: