# Basis of a Vector Space Example

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

This sample will help you deal with the basis of a vector sector. You will spend much less time on your assignment with the help of the sample. Of course, this basis of a vector sector example was created by an expert who is knowledgeable in the topic. Therefore, you can be sure that it is absolutely correct.

A lot of students have already used AssignmentShark and have received the best possible help with their assignments. You can use our service as well, and we will help you in the fastest possible time. Add your requirements in the order form and we will help you get the highest grade for your homework.