Row and Column Vectors#

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:

\[ \begin{bmatrix}v_1 & \cdots & v_m\end{bmatrix} \in F^{1\times n} \]

Bra-Notation

Sometimes, a row vector \(\mathbf{v}\) can be written as \(\left\langle v \right\vert\). This is known as bra-notation.

Definition: Column Vector

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

\[ \begin{bmatrix}v_1 \\ \vdots \\ v_m\end{bmatrix} \in F^{m\times 1} \]

Ket-Notation

Sometimes, a column vector \(\mathbf{v}\) can be written as \(\left\vert v \right\rangle\). 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

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

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

Proof

TODO

Definition: Standard Basis

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