Vector Space
Note: Text in italics are comments about the non-italic parts usually above.
By a vector space we mean a nonempty set $E$ with two operations:
This just means there is some collection of objects, called $E$, and it is not empty. Why “nonempty”? Because a vector space must at least have something in it. In fact, the rules will force it to contain a special element called the zero vector. The elements of $E$ are called vectors. Important: at this stage, “vector” does not mean “arrow in space.” It just means “element of this set $E$.” It could be: arrows, tuples like $(𝑥,𝑦,𝑧) (x,y,z)$, functions, polynomials, matrices, sequences, and so on. So “vector” here means “thing that behaves according to these rules.”
\((x, y) \mapsto x + y\) from \(E \times E\) into \(E\), called addition,
This says: Take any two vectors $𝑥$ and $𝑦$ from $𝐸$, and there is a rule that produces another element $𝑥+𝑦$, also in $𝐸$. What does "from $𝐸×𝐸$ into $𝐸$" mean? $E×E$ means all ordered pairs $(𝑥,𝑦)$ with both $𝑥$, $𝑦$ $∈$ $𝐸$. "into $𝐸$" means the result is again an element of $𝐸$.
So addition is a function:
input: two vectors,
output: one vector.
This is the closure idea: adding vectors keeps you inside the space
$(\lambda, x) \mapsto \lambda x$ from $\mathbb{F} \times E$ into $E$, called multiplication by scalars,
This says: Take a scalar $𝜆$ from $𝐹$, and a vector $𝑥$ from $𝐸$, and the rule gives another vector $𝜆𝑥$ in $𝐸$. Here $𝐹$ is the field of scalars, $F=R$ for real numbers, or $𝐹=𝐶$ for complex numbers.
So scalar multiplication means: numbers can “act on” vectors.
Again, the result must stay in $𝐸$.
Big picture so far
A vector space is: a set $𝐸$ of vectors, an addition rule for vectors, a scalar multiplication rule using numbers from $𝐹$, with some laws (coming up). Those laws are what make the structure behave like ordinary linear geometry.
Such that the following conditions are satisfied for all $x, y, z \in E$ and $\alpha, \beta \in \mathbb{F}$:
$x + y = y + x$;
This is commutativity of addition. It says the order of addition does not matter. You want vector addition to behave symmetrically. No vector should get privileged treatment just because it was written first.
By a vector space we mean a nonempty set $E$ with two operations:
- $(x, y) \mapsto x + y$ from $E \times E$ into $E$, called addition,
- $(\lambda, x) \mapsto \lambda x$ from $\mathbb{F} \times E$ into $E$, called multiplication by scalars,
such that the following conditions are satisfied for all $x, y, z \in E$ and $\alpha, \beta \in \mathbb{F}$:
- $x + y = y + x$;
- $(x + y) + z = x + (y + z)$;
- For every $x, y \in E$ there exists a $z \in E$ such that $x + z = y$;
- $\alpha(\beta x) = (\alpha \beta)x$;
- $(\alpha + \beta)x = \alpha x + \beta x$;
- $\alpha(x + y) = \alpha x + \alpha y$;
- $1x = x$.
Elements of $E$ are called vectors. If $\mathbb{F} = \mathbb{R}$, then $E$ is called a real vector space, and if $\mathbb{F} = \mathbb{C}$, $E$ is called a complex vector space.