A basis β for a vector space V is a linearly independent subset of V
that generates
or span V. If β is a basis for V, we also say that elements of β form a basis for
V. This means that every vector from V is a finite linear combination of elements from the basis.

Recall that a set of vectors β is said to generate or span a vector space V if every element from
V can be represented as a linear combination of vectors from β.

Example: The span of the empty set \( \varnothing \) consists of a unique element 0.
Therefore, \( \varnothing \) is linearly independent and it is a basis for the trivial vector space
consisting of the unique element---zero. Its dimension is zero.

Example: In \( \mathbb{R}^n , \) the vectors
\( e_1 [1,0,0,\ldots , 0] , \quad e_2 =[0,1,0,\ldots , 0], \quad \ldots , e_n =[0,0,\ldots , 0,1] \)
form a basis for n-dimensional real space, and it is called the standard basis. Its dimension is n.

Example: Let us consider the set of all real \( m \times n \)
matrices, and let \( {\bf M}_{i,j} \) denote the matrix whose only nonzero entry is a 1 in
the i-th row and j-th column. Then the set \( {\bf M}_{i,j} \ : \ 1 \le i \le m , \ 1 \le j \le n \)
is a basis for the set of all such real matrices. Its dimension is mn.

Example: The set of monomials \( \left\{ 1, x, x^2 , \ldots , x^n \right\} \)
form a basis in the set of all polynomials of degree up to n. It has dimension n+1.
■

Example: The infinite set of monomials \( \left\{ 1, x, x^2 , \ldots , x^n , \ldots \right\} \)
form a basis in the set of all polynomials.
■

Theorem: Let V be a vector space and
\( \beta = \left\{ {\bf u}_1 , {\bf u}_2 , \ldots , {\bf u}_n \right\} \) be a subset of
V. Then β is a basis for V if and only if each vector v in V can be uniquely
decomposed into a linear combination of vectors in β, that is, can be uniquely expressed in the form