Bases and Dimension

Links: Linear Independence, Linear Combinations

Definition Basis

A basis $β$ for a vector space $V$ is a linearly independent subset of $V$ that generates V. If $β$ is a basis for V, we also say that the vectors of $β$ form a basis for V.

Theorems Basis

Theorem: Let $β = {u_{1}, u_{2}, \dots, u_{n}}$ is a basis for $V$ if and only if each $v \in V$ can be uniquely expressed as a linear combination of vectors of $β$ , that is, can be expressed in the form:

v = \sum_{i = 1}^{n} a_{i} u_{i}

for unique scalars $a_{1}, a_{2}, \dots a_{n}$ .

Theorem: Let a vector space $V$ is generated by a finite set $S$ , then some subset of $S$ is a basis for $V$ . Hence $V$ base a finite basis.

Replacement Theorem: Let $V$ be a vector space that is generated by a set $G$ containing exactly $n$ vectors. and let $L$ be a linearly independent subset of $V$ containing exactly $m$ vectors. Then $m \leq n$ and there exists a subset $H$ of $G$ containing exactly $n - m$ vectors such that $L \cup H$ generates $V .$

Corollary: Let $V$ be a vector space having a finite basis. Then all bases for $V$ are finite, and every basis for $V$ contains the same number of vectors.

Definition: A vector space is called finite-dimensional if it has a basis consisting of a finite number of vectors. The unique integer $n$ such that every basis for $V$ contains exactly $n$ elements is called the dimension of $V$ and is denoted by $dim (V)$ . A vector space that is not finite-dimensional is called infinite-dimensional.

Theorems Dimension

Corollary: Let V be a vector space with dimension n:

Any finite generating set for $V$ contains at least $n$ vectors, and a generating set for $V$ that contains exactly $n$ vectors is a basis for $V$ .
Any linearly independent subset of $V$ that contains exactly $n$ vectors is a basis for $V .$
Every linearly independent subset of $V$ can be extended to a basis for $V$ , that is, if $L$ is a linearly independent subset of $V$ , then there is a basis $β$ of $V$ such that $L \sube β .$

Theorem: Let $W$ be a subspace of a finite-dimensional vector space $V$ . Then $W$ is finite-dimensional and $dim (W) \leq dim (V)$ . Moreover, if $dim (W) = dim (V)$ , then $V = W$ .

Corollary: If $W$ is a subspace of a finite-dimensional vector space $V$ , then any basis for $W$ can be extended to a basis for $V .$

Let ${U_{i}}_{i = 0}^{n}$ be a collection of subspaces of $V$ a finite dimensional vector space, such that $\sum U_{i}$ is a direct sum $(⨁ U_{i})$ , iff:

\dim (⨁_{i = 0}^{n} U_{i}) = \sum_{i = 0}^{n} \dim (U_{i})