Vector bases and dimension

Overview

Every vector space has a basis, which combines notions from spans and linear independence. We can study vector spaces by studying their bases.

One special basis is the standard basis for $F^n$ (i.e., the $n$-fold product of a field $F$), which is the set $\{e_1,\cdots ,e_n\}$ where the only non-zero coordinate in each vector $e_i$ is the $i$th coordinate. Another example is $\{1,x,\dots ,x^n\}$, which is a basis for the vector subspace $P_n$ of the polynomial ring $F[x]$.

Dimensions tell us exactly how many coordinates are needed to describe an arbitrary element of a vector space.

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Bases

Definition: Basis of a vector space

If $V$ is a finite-dimensional vector space, a basis for $V$ is a set of vectors $\{v_1,\cdots ,v_n\}\subset V$ which:

• (i) Spans the entire space, so every vector in $V$ is some linear combination of these vectors;
• (ii) Is linearly independent, so only the trivial linear combination produces $\vec{0}$.

Definition: Standard basis of a field

If $F$ is a field, the standard basis of the $n$-fold Cartesian product $F^n$ is the sequence $\{e_1,\dots ,e_n\}$, where the only nonzero coordinate in each $e_i$ is the $i$th coordinate.

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Counting and dimension

Lemma: Main counting arguemnt

If $w_1,\dots ,w_b$ are linearly independent vectors all contained in the span $\text{span}\{v_1,\dots ,v_a\}$ then $b\le a$.

Corollary: Dimension of a vector space

Let $V$ be a finite-dimensional $F$-vector space, meaning there exist $v_1,\dots ,v_d\in V$ such that $V=\text{span}\{v_1,\dots ,v_d\}$.

• (i) Dimension is well-defined: Any two bases for $V$ have the same number of elements. In this case, the number of elements in the basis is called the dimension of $V$, written ${\dim }_{F}V$.
• (ii) Basis reduction: If $V=\text{span}\{v_1,\dots ,v_d\}$, then there is a subsequence of $v_1,\dots ,v_d$ that is a basis for $V$. Hence $\dim V\le d$, with equality if and only if the entire sequence is a basis for $V$.
• (iii) Basis extension: If $w_1,\dots ,w_{\ell }\in V$ are linearly independent, then there exist vectors $w_{\ell +1},\dots ,w_r\in V$ such that $w_1,\dots ,w_{\ell },w_{\ell +1},\dots ,w_r$ is a basis for $V$. Hence $\dim V\ge \ell$.
• (iv) If $W\subseteq V$ is a vector subspace, then $\dim W\le \dim V$, with equality if and only if $W=V$.
• (v) If $v_1,\dots ,v_d$ is a basis for $V$, then the map $f:F^d\to V$ defined by $f(t_1,\dots ,t_d)={\sum }_{i=1}^d{t}_i{v}_i$ is a linear isomorphism.