Bases

Definition

Definition 1(Definition).

A system of vectors ${\mathbf{v}}_1,{\mathbf{v}}_2,{\mathbf{v}}_3,\cdots {\mathbf{v}}_n\in V$, where $V$ is a vector space, is called a basis of $V$ if any $\mathbf{v}\in V$ can be uniquely expressed as a linear combination of the system.

Note

Given a vector space $V$ and a set of basis vectors, any vector in $V$ is uniquely determined by the coefficients in its decomposition in terms of the basis vectors. So, we can stack the coefficients in a column and operate on them like we do with elements of ${\mathbb{F}}^n$. This allows us to translate any statement about vectors in ${\mathbb{F}}^n$ to a vector space $V$ with a given basis.

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Building up to a basis

The existence and uniqueness of linear combinations requirements of bases can be analyzed separately.

Spanning systems

Definition 2(Definition).

A system of vectors ${\mathbf{v}}_1,{\mathbf{v}}_2,\dots ,{\mathbf{v}}_p\in V$ is called a spanning (aka complete, generating) system in $V$ if any vector $\mathbf{v}\in V$ admits a representation as a linear combination of the vectors.

This differs from a basis in that it does not require the representations to be unique.

Linearly independent systems

Definition 3(Definition).

A system of vectors ${\mathbf{v}}_1,{\mathbf{v}}_2,\dots ,{\mathbf{v}}_p\in V$ is called linearly independent iff the equation $x_1{\mathbf{v}}_1+x_2{\mathbf{v}}_2+\cdots +x_p{\mathbf{v}}_p=0$ (with unknowns $x_k$) has only trivial solution.

By definition, a basis is a linearly independent system.

A system of vectors is called linearly dependent if $0$ can be expressed as a non-trivial linear combination of the vectors.

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New, equivalent definition of basis

It is clear from the definition of a basis that a basis is both spanning and linearly independent. It can also be proved (LADW, p10) that if a set is spanning and linearly independent, it is a basis. Thus, A system of vectors is a basis if and only if it is a spanning system and linearly independent.

Any (finite) generating system contains a basis.

Suppose ${\mathbf{v}}_1,{\mathbf{v}}_2,\dots ,{\mathbf{v}}_p\in V$ is a generating set. If it is linearly independent, we are done. If not, then there exists a vector $v_k$ which can be expressed as a linear combination of the vectors ${\mathbf{v}}_{j\ne k}$. So, any linear combination of the vectors ${\mathbf{v}}_1,{\mathbf{v}}_2,\dots ,{\mathbf{v}}_p$ can be represented as a linear combination of the same vectors without $v_k$(just replace $v_k$ with its expression as a linear combination of the other vectors). Thus, $v_k$ can be removed from the set while retaining the spanning property of the set. If we keep repeatedly removing such vectors from the set, we must end up with a basis (we cannot reach the null set as we are preserving completeness on every removal).

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Standard basis in ${\mathbb{F}}^n$

${\mathbf{e}}_1:=\left(\begin{array}{c}1\\ 0\\ 0\\ ⋮\\ 0\end{array}\right),{\mathbf{e}}_2:=\left(\begin{array}{c}0\\ 1\\ 0\\ ⋮\\ 0\end{array}\right),{\mathbf{e}}_3:=\left(\begin{array}{c}0\\ 0\\ 1\\ ⋮\\ 0\end{array}\right),\dots ,{\mathbf{e}}_n:=\left(\begin{array}{c}0\\ 0\\ 0\\ ⋮\\ 1\end{array}\right)$

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Standard basis in ${\mathbb{P}}^n$

${\mathbf{e}}_0:=1,{\mathbf{e}}_1:=t,{\mathbf{e}}_2:=t^2,{\mathbf{e}}_3:=t^3,\dots ,{\mathbf{e}}_n:=t^n$