A set of vectors in vector space V is called a basis for V if the vectors in the set are linearly independent and every vector is a linear combination of vectors in this set. The former one equals the vector space being spanned by the set. The dimension of vector space V is the number of vectors in its basis. A vector space is called nite dimensional if it has a nite basis.
Example 1. Let be the real vector space in three dimensions, and consider the set of vectors S=; ; g where = (1; 2; 0), = (0; 1; 0) and = (1; 0; 1). Let us see whether S is a basis for or not.
S is linearly independent since the solution for the equation ++ = 0 is unique, and it is
On the other hand, we need to investigate whether the vectors span the whole vector space. Any vector in is of the form (a; b; c). We need to end coefficients and in terms a, b and c.
Once we apply Gauss-Jordan reduction to this equation we get the following solution:
The standard basis for is f(1; 0; 0); (0; 1; 0); (0; 0; 1)g. The dimension of is three, since there are three vectors in its bases.
Example 2. Now, consider the set of vectors ; where and and consider another set of vectors where and
The first set is not a basis for since it has two vectors, and a set of two vectors cannot span a vector space with dimension three. The second set is not a basis for because it is not linearly independent, since
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