Row and Column Vectors

Introduction

Two types of matrices are very important for most of mathematics and physics.

Definition: Row Vector

An $n$-dimensional row vector is an $1\times n$-matrix:

$$\left[\begin{array}{ccc}{v}_1& \cdots & v_m\end{array}\right]\in F^{1\times n}$$

Bra-Notation

Sometimes, a row vector $\mathbf{v}$ can be written as $⟨v|$. This is known as bra-notation.

Definition: Column Vector

An $m$-dimensional column vector is an $m\times 1$-matrix:

$$\left[\begin{array}{c}{v}_1\\ ⋮\\ v_m\end{array}\right]\in F^{m\times 1}$$

Ket-Notation

Sometimes, a column vector $\mathbf{v}$ can be written as $|v⟩$. This is known as ket-notation.

NOTATION

We usually denote the set of all $m$-dimensional column vectors by $F^m$ instead of $F^{m\times 1}$.

NOTATION

It is very common to denote row and column vectors with an arrow above a lowercase letter: $\vec{u},\vec{v}$, etc.

Theorem: Standard Basis

If $(F^n,F,+,\cdot )$ is the vector space of the $n$-dimensional column vectors over some field $F$, then the $n$-tuple

$$\left(\underset{n\text{ vectors}}{\underbrace{{\vec{e}}_1=\left[\begin{array}{c}{1}_F\\ 0_F\\ ⋮\\ 0_F\end{array}\right],{\vec{e}}_2=\left[\begin{array}{c}{0}_F\\ 1_F\\ ⋮\\ 0_F\end{array}\right],\cdots ,{\vec{e}}_n=\left[\begin{array}{c}{0}_F\\ 0_F\\ ⋮\\ 1_F\end{array}\right]}}\right)$$

is an ordered basis for $(F^n,F,+,\cdot )$.

PROOF

TODO

Definition: Standard Basis

This ordered basis is known as the standard basis of $(F^n,F,+,\cdot )$.