Coordinates Relative to a Basis

If the basis of a vector space is known, then every vector (element of the vector space) can be expressed as a linear combination of the basis vectors.

This is to be expected, since the basis of a vector space contains only linearly independent vectors, so any non-basis vectors must be linearly dependent, and hence able to be expressed using a linear combination.

So far, it hasn’t been properly mentioned, but almost all references to bases refer to ordered bases, where elements of the bases have a certain order and can’t be rearranged once that order is defined.

Expressing vectors from a basis uniquely ^theorem

If $\mathcal{B}=\{{\vec{v}}_1,{\vec{v}}_2,{\vec{v}}_3,\dots ,{\vec{v}}_k\}$ is the ordered basis of a vector space (fixed over the field $\mathbb{F}$)$V_{\mathbb{F}}$ then every vector in $V$ can be expressed uniquely, as a linear combination of the basis, i.e.:

$$\vec{v}=a_1{\vec{v}}_1+a_2{\vec{v}}_2+a_3{\vec{v}}_3+\dots a_k{\vec{v}}_k{a}_{1\dots k}\in \mathbb{F}$$

This is usually done by solving a linear system consisting of the basis vectors.

Now that we know every vector can be expressed, we can connect vector spaces to a coordinate system (such as that of the Cartesian Plane ($x,y$)), using coordinate vectors

Coordinate Vectors §

Coordinates relative to a Basis

If $\mathcal{B}=\{{\vec{v}}_1,{\vec{v}}_2,{\vec{v}}_3,\dots ,{\vec{v}}_k\}$ is the ordered basis of a vector space (fixed over the field $\mathbb{F}$)$V_{\mathbb{F}}$ then every vector in $V$ can be expressed uniquely as:

$$\vec{v}=a_1{\vec{v}}_1+a_2{\vec{v}}_2+a_3{\vec{v}}_3+\dots a_k{\vec{v}}_k{a}_{1\dots k}\in \mathbb{F}$$

The set of unique scalars, $\{a_1,a_2,\dots ,a_k\}$ are the coordinates of $\vec{v}$ relative to $\mathcal{B}$. Then:

$$[\vec{v}{]}_{\mathcal{B}}=\left[\begin{array}{c}{a}_1\\ a_2\\ a_3\\ \dots \\ a_k\end{array}\right]$$

is the coordinate vector of $\vec{v}$ with respect to $\mathcal{B}$

The existence of these coordinate vectors means that a vector in $V_{\mathbb{F}}$ can be uniquely represented by any basis of $V_{\mathbb{F}}$.

Furthermore, these coordinate vectors are always linked, even during a linear combination:

Coordinate Vectors are Vectors

Let $V_{\mathbb{F}}$ be a vector space fixed with the field $\mathbb{F}$. Also, let $\mathcal{B}$ be the ordered basis of $V_{\mathbb{F}}$. If $\vec{u},\vec{v}\in V_{\mathbb{F}}$ and $k\in \mathbb{F}$:

$$[\vec{u}+\vec{v}{]}_{\mathcal{B}}=[\vec{u}{]}_{\mathcal{B}}+[\vec{v}{]}_{\mathcal{B}}$$

$$[k\vec{u}{]}_{\mathcal{B}}=k[\vec{u}{]}_{\mathcal{B}}$$