Properties of Dimension

In the setting of Linear Algebra: Standing Notation, let $K$, with $+$, $\cdot$, $0$ and $1$, be a field, let $V$, with its addition and scalar multiplication, be a finite-dimensional vector space over $K$ with dimension $\dim V$, let $m\in\mathbb{N}_{0}$, and let $u=(u_{1},\dots,u_{m})$ be an $m$-tuple in $V$. Linear independence is as in Linearly Independent and Linearly Dependent Finite Families (i), spanning as in Span of a Finite Family; Spanning Families (iii), and bases as in Finite Basis of a Vector Space (i).

(i) Extension to a basis. If $u$ is linearly independent, then there are $p\in\mathbb{N}_{0}$ and a $p$-tuple $(w_{1},\dots,w_{p})$ in $V$ such that

$$(u_{1},\dots,u_{m},w_{1},\dots,w_{p}),$$

the $(m+p)$-tuple whose $i$-th component is $u_{i}$ for $i\in[m]$ and $w_{i-m}$ for $i\in\{m+1,\dots,m+p\}$, is a basis of $V$.

(ii) Independent families. If $u$ is linearly independent, then $m\leq\dim V$, and $m=\dim V$ if and only if $u$ is a basis of $V$.

(iii) Spanning families. If $u$ spans $V$, then $m\geq\dim V$, and $m=\dim V$ if and only if $u$ is a basis of $V$.

(iv) Dimension zero. $\dim V=0$ if and only if $V=\{0\}$.

(v) Subspaces. Let $W$ be a linear subspace of $V$, a vector space over $K$ with the operations of $V$ by Linear Subspaces Are Vector Spaces; Intersections (i). Then $W$ is finite-dimensional, $\dim W\leq\dim V$, and $\dim W=\dim V$ if and only if $W=V$.

(vi) Euclidean space. Let $n\in\mathbb{N}$. In the real vector space $\mathbb{R}^{n}$ of ℝⁿ Is a Real Vector Space (i), the standard basis vectors $(e_{1},\dots,e_{n})$ form a basis, and

$$\dim\mathbb{R}^{n}=n.$$