Basis

Definition: Basis

A basis of $V$ is a list of vectors in $V$ that is linearly independent and spans $V$.

Examples:

• The list $(1,0,\dots ,0),(0,1,0,\dots ,0),\dots (0,\dots ,0,1)$ is a basis of ${\mathbb{F}}^n$, called the standard basis of ${\mathbb{F}}^n$.
• The list $(1,2),(3,5)$ is a basis of ${\mathbb{F}}^2$. Note that this list has length 2, which is the same as the length of the standard basis of ${\mathbb{F}}^2$.
• The list $(1,2,-4),(7,-5,6)$ is linearly independent in ${\mathbb{F}}^3$ but is not a basis of ${\mathbb{F}}^3$ because it does not span ${\mathbb{F}}^3$.
• The list $(1,2),(3,5),(4,13)$ spans ${\mathbb{F}}^3$ but is not a basis of ${\mathbb{F}}^2$ because it is not linearly independent.
• The list $(1,1,0),(0,0,1)$ is a basis of $\{(x,x,y)\in {\mathbb{F}}^3:x,y,\in \mathbb{F}\}$
• The list $(1,-1,0),(1,0,-1)$ is a basis of $\{(x,y,z)\in {\mathbb{F}}^3:x+y+z=0\}$
• The list $1,z,\dots ,z^m$ is a basis of ${\mathcal{P}}_m(\mathbb{F})$, called the standard basis of ${\mathcal{P}}_m(\mathbb{F})$.

Criterion for Basis

Criterion for Basis

A list $v_1,\dots ,v_n$ of vectors in $V$ is a basis of $V$ if and only if every $v\in V$ can be written uniquely in the form

$$v=a_1{v}_1+\cdots +a_n{v}_n$$

where $a_1,\dots ,a_n\in \mathbb{F}$.

Proof. First suppose that $v_1,\dots ,v_n$ is a basis of $V$. Let $v\in V$. Because $v_1,\dots ,v_n$ spans $V$, there exist $a_1,\dots a_n\in \mathbb{F}$ such that $v=a_1{v}_1+\cdots +a_n{v}_n$ holds. To show that the representation is unique, suppose that we also have

$$v=c_1{v}_1+\cdots +c_n{v}_n$$

Subtracting the two equations from each other would give us

$$0=(a_1-c_1)v_1+\cdots +(a_n-c_n)v_n$$

This implies that each $a_k-c_k$ equals $0$, because $v_1,\dots ,v_n$ is linearly independent. Hence, $a_1=c_1,\dots ,a_n=c_n$. We have the desired uniqueness. This completes the proof in one direction.

From the other direction, suppose every $v\in V$ can be written uniquely in the form of $v=a_1{v}_1+\cdots +a_n{v}_n$. This means that the list $v_1,\dots ,v_n$ spans $V$. To show that $v_1,\dots ,v_n$ is linearly independent, suppose $a_1,\dots ,a_n\in \mathbb{F}$ such that

$$0=a_1{v}_1+\cdots +a_n{v}_n$$

The uniqueness of the representation implies that $a_1=\cdots =a_n=0$. If there were any solutions other than the trivial one, they would not be unique (we would have other nonzero combinations summing to $0$). Thus $v_1,\dots ,v_n$ is linearly independent and hence is the basis of $V$.