Vector

linear-algebra

Definition

Vector

A vector $v\in V$, where $V$ is a vector space, is a quantity that has magnitude and direction and is of form:

$$\mathbf{v}=\underset{\text{Column Vector}}{\underbrace{\left[\begin{array}{c}{\mathbf{v}}_1\\ {\mathbf{v}}_2\\ ⋮\\ {\mathbf{v}}_n\end{array}\right]}}=\underset{\text{Row Vector}}{\underbrace{\left[\begin{array}{cccc}{\mathbf{v}}_1& {\mathbf{v}}_2& \dots & {\mathbf{v}}_n\end{array}\right]}}$$

where ${\mathbf{v}}_i\in F$ with $F$ being the scalar field of the vector space $V$.

Arithmetic

Addition

Let $a,b\in V$, where $V$ is a vector space, then:

$$\mathbf{a}+\mathbf{b}=\left[\begin{array}{c}{\mathbf{a}}_1\\ {\mathbf{a}}_2\\ ⋮\\ {\mathbf{a}}_n\end{array}\right]+\left[\begin{array}{c}{\mathbf{b}}_1\\ {\mathbf{b}}_2\\ ⋮\\ {\mathbf{b}}_n\end{array}\right]=\left[\begin{array}{c}{\mathbf{a}}_1+{\mathbf{b}}_1\\ {\mathbf{a}}_2+{\mathbf{b}}_2\\ ⋮\\ {\mathbf{a}}_n+{\mathbf{b}}_n\end{array}\right]$$

Using the parallelogram law, we can visualise the addition of two vectors as the following:

Subtraction

Let $a,b\in V$, where $V$ is a vector space, then:

$$\mathbf{a}-\mathbf{b}=\left[\begin{array}{c}{\mathbf{a}}_1\\ {\mathbf{a}}_2\\ ⋮\\ {\mathbf{a}}_n\end{array}\right]-\left[\begin{array}{c}{\mathbf{b}}_1\\ {\mathbf{b}}_2\\ ⋮\\ {\mathbf{b}}_n\end{array}\right]=\left[\begin{array}{c}{\mathbf{a}}_1-{\mathbf{b}}_1\\ {\mathbf{a}}_2-{\mathbf{b}}_2\\ ⋮\\ {\mathbf{a}}_n-{\mathbf{b}}_n\end{array}\right]$$

Scalar Multiplication

Multiplication mit Skalar

Let $a,b\in V$ and $\lambda \in F$, where $V$ is a vector space and $F$ be the scalar field of $V$, then:

$$\lambda \cdot \mathbf{a}=\lambda \cdot \left[\begin{array}{c}{\mathbf{a}}_1\\ {\mathbf{a}}_2\\ ⋮\\ {\mathbf{a}}_n\end{array}\right]=\left[\begin{array}{c}\lambda \cdot {\mathbf{a}}_1\\ \lambda \cdot {\mathbf{a}}_2\\ ⋮\\ \lambda \cdot {\mathbf{a}}_n\end{array}\right]$$

Dot Product

Definition

Inner Product

An inner product is a function that takes in two vectors of a vector space $V$ and outputs an element of a scalar field $\mathbb{F}$ ($\mathbb{F}$ being either $\mathbb{R}$ or $\mathbb{C}$):

$$⟨\cdot ,\cdot ⟩:V\times V\to \mathbb{F}$$

with the following properties:

1. Conjugate symmetry:

$$⟨u,v⟩=\overline{⟨v,u⟩}\forall u,v\in V$$

1. Sesquilinearity: The operation must respect linearity in the first argument, i.e.:

$$⟨cu+v,w⟩=c⟨u,w⟩+⟨v,w⟩\forall u,v,w\in V,c\in \mathbb{F}$$

1. Positive definiteness: The inner product of a vector with itself must be non-negative, and it is zero iff the vector itself is the zero vector, i.e.:

$$\begin{array}{rl}⟨v,v⟩& \ge 0\forall v\in V\\ ⟨v,v⟩& =0⟺v=0\forall v\in V\end{array}$$

Link to original

Cross Product

Kreuzprodukt

$$\mathbf{a}\times \mathbf{b}=\left[\begin{array}{c}{\mathbf{a}}_1\\ {\mathbf{a}}_2\\ {\mathbf{a}}_3\end{array}\right]\times \left[\begin{array}{c}{\mathbf{b}}_1\\ {\mathbf{b}}_2\\ {\mathbf{b}}_3\end{array}\right]=\underset{\text{Symmetric Pattern}}{\underbrace{\left[\begin{array}{c}{\mathbf{a}}_2{\mathbf{b}}_3-{\mathbf{a}}_3{\mathbf{b}}_2\\ {\mathbf{a}}_3{\mathbf{b}}_1-{\mathbf{a}}_1{\mathbf{b}}_3\\ {\mathbf{a}}_1{\mathbf{b}}_2-{\mathbf{a}}_2{\mathbf{b}}_1\end{array}\right]}}$$

The resulting vector $\mathbf{a}\times \mathbf{b}$ is orthogonal on $\mathbf{a}$ and $\mathbf{b}$: