11.1 Vector space

• A vector space consists of the following four elements:
  • A set of vectors $V$ ,
  • A field of scalars $$ (where, for our purposes, $$ is either $\mathbb{R}$ or $\mathbb{C}$ ),
  • The operations of vector addition "+" ( i.e. , + : $V\times V\to V$ )
  • The operation of scalar multiplication "⋅"( i.e. ,⋅: $\times V\to V$ )
  for which the following properties hold. (Assume $(x\land y\land z)\in V$ and $(\alpha \land \beta )\in $ .)

Important examples of vector spaces include

• A subspace of $V$ is a subset $M\subset V$ for which
  • $\forall x, y, x\in M\land y\in M\colon (x+y)\in M$
  • $\forall , x\in M\land \alpha \in \colon \alpha x\in M$
  Note that every subspace must contain $0$ , and that $V$ is a subspace of itself.
• The span of set $S\subset V$ is the subspace of $V$ containing all linear combinations of vectors in $S$ . When $S=\{x_0, \dots , x_{N-1}\}$ , $$≔(\mathrm{span}(S), \{\sum_{i=0}^{N-1} {\alpha }_i{x}_i\colon {\alpha }_i\in \})$$

• A subset of linearly-independent vectors $\{x_0, \dots , x_{N-1}\}\subset V$ is called a basis for V when its span equals $V$ . In such a case, we say that $V$ has dimension $N$ . We say that $V$ is infinite-dimensional
  The definition of an infinite-dimensional basis would becomplicated by issues relating to the convergence of infinite series. Hence we postpone discussion of infinite-dimensionalbases until the Hilbert Space section.
  if it contains an infinite number of linearly independent vectors.
• $V$ is a direct sum of two subspaces $M$ and $N$ , written $V=M\mathop{\mathrm{xor}}N$ , iff every $x\in V$ has a unique representation $x=m+n$ for $m\in M$ and $n\in N$ .
  Note that this requires $M\cap N=\{0\}$