Vectors & Unit vectors

Vectors

Basic vector

• collection of numbers, arranged in col or row
• can be thought of as the coordinates of a point in n-dimensional space
• usually written in column format: $x=\left[\begin{array}{c}{x}_1\\ x_2\\ ⋮\\ x_n\end{array}\right]$
• $x_i$: i-th component of vector x
• $x_n$: n dimension of x
• ${\mathbb{R}}^n$: set of n-vectors (vectors of dimension n)

Transpose

• transpose: an operation to transform column vector x in row format and vice versa, denoted by ${}^T$ (superscript T)

function note:

• x, y both unit-norm: $(x-y{)}^{\top }(x+y)=x^{\top }x-y^{\top }y-y^{\top }x+x^{\top }y=x^{\top }x-y^{\top }y=0$

Unit vectors

• the $i-th$ unit vector is the n-vector $e_i$ with all entries = 0 except the i-th one $e_2=\left[\begin{array}{c}0\\ 1\\ 0\\ 0\end{array}\right]$