Bases

A basis for a vector space $V$ is a linearly independent subset of $V$ that generates $V$

$\beta$ is a basis for $V$ iff $V$ can be uniquely expressed as a linear combination of vectors of $\beta$ (theorem 1.8)

If $V$ generated by finite set $S$, then some subset of $S$ is a basis for $V$ (so $V$ has a finite basis) (theorem 1.9) → A finite spanning set for $V$ can be reduced to a basis for $V$

Dimensions

• A vector space is called finite-dimensional if it has a finite basis
• The unique number of vectors in each basis is called the dimension
• Notated $\dim (V)$

Replacement Theorem

• Definition
  • $V$ generated by a set $G$ containing $n$ vectors
  • $L$ is a linearly independent subset of $V$ containing $m$ vectors
  • Then $m\le n$ and there exists a subset $H\subseteq G$ containing exactly $n-m$ vectors such that $L\cup H$ generates $V$
• Facts
  • No linearly independent subset of $V$ can contain more than $\dim (V)$ vectors
• Corollaries
  1. Let $V$ be a vector space having a finite basis. Then every basis for V contains the same number of vectors
  2. Let $\dim (V)=n$
     • Any finite generating set for $V$ contains at least $n$ vectors. A generating set with $n$ vectors is a basis
     • Any linearly independent subset of $V$ with $n$ vectors is a basis for $V$
     • Every linearly independent subset of $V$ can be extended to be a basis for $V$

Relationship with Subspaces

• Let $W$ be a subspace of a finite-dimensional vector space $V$
• Then $W$ is also finite-dimensional and $\dim (W)\le \dim (V)$
• If $\dim (W)=\dim (V)$, then $W=V$
• Any basis for $W$ can be extended to the basis for $V$

Standard Ordered Basis

• Ordered basis: a basis with a specific order
• Examples
  • $F^n:\{e_1,\dots ,e_n\}$
  • $P_n(F):\{1,x,x^2,\dots ,x^n\}$

Coordinate Vectors

• For $x\in V$, the coordinate vector is $x$ relative to $\beta$
• Denoted $[x{]}_{\beta }$

$$\begin{array}{rl}\beta & =\{u_1,\dots ,u_n\}\\ x& =\sum _{i=1}^n{a}_i{u}_i\\ & \\ [x{]}_{\beta }& =\left(\begin{array}{c}{a}_1\\ a_2\\ ⋮\\ a_n\end{array}\right)\end{array}$$