Coordinates, Standard Basis and Dimension

Coordinates and Standard Basis

Coordinate: Express $w$ in terms of a basis $S=\{v_1,v_2,\dots ,v_n\}$ from a vector $v$ on a field $\mathbb{R}$ (or $\mathbb{C}$):

$w=k_1{v}_1+k_2{v}_2+\cdots +k_n{v}_n$

The unique vector $(k_1,k_2,\dots ,k_n)$ formed from the scalar coefficients in ${\mathbb{R}}^n$ (or ${\mathbb{C}}^n$) is said to be the coordinate vector or simply the coordinate of $w$ relative to $S$, denoted as

$(w{)}_S=(k_1,k_2,\dots ,k_n)$

the standard basis:

$e_1=(1,0,\dots ,0),e_2=(0,1,\dots ,0)$ $e_n=(0,0,\dots ,1)$

where each $e_i$ has the $i$th component expressed as one and all others as zero. Uniquely expressed as

$v=v_1{e}_1+v_2{e}_2+\cdots +e_n{v}_n={\sum }_{i=1}^n{v}_i{e}_i$

$\therefore \{e_1,e_2,\dots ,e_n\}$ is the standard basis of ${\mathbb{R}}^n$ (or ${\mathbb{C}}^n$). By default, the coordinate of a vector is relative to the standard basis:

$⟨v_1,v_2,\dots ,v_n⟩(e_1,e_2,\dots ,e_n)=⟨v_1,v_2,\dots ,v_n⟩$

Dimension

Intuitively, we can reason about lines (1D), planes (2D) and 3D spaces. In linear algebra, dimension for finite-dimensional vector space is defined as:

For a finite-dimensional vector space, all bases posses the same number of vectors.

Dimension: number of vectors in a basis, denoted $\dim (V)$ where $V\Rightarrow$ vector space.

Subspaces: lower dimension constructs of a vector space.