Meanwhile, if V is some vector space having a basis consisting of 'n' elements, we say that n is the dimension of V. If V consists of 0 alone, then V does not have a basis, and one says V has dimension 0.

As another generic example, let the vector space

**R**^n have dimension n over

**R**, and the vector space

**C**^n have dimension n over

**C**. More generally, for any field F, the vector space F^n has dimension n over F. One can therefore say that the n vectors:

(1, 0.........), (0,1...........0), (0, 0, 1..........0),.........(0, ............0, 1)

form a basis of F^n over F.

Then the dimension of the vector space V over F is denoted by: dim_F V or more simply, dim V.

Associated with these considerations is the concept of a

**maximal set of linearly independent elements**of a vector space. Then let: v1, v2, v3.......vn be linearly independent elements of a vector space V, then the elements w, v1, v2, .....vn are linearly independent and v1, v2, v3.......vn for a set of maximally independent elements.

Ancillary or Auxilliary Theorems:

1) Given V is a vector space and one basis has m elements and another basis has n, then m = n.

2) Let V be a vector space and {v1, v2, v3.......vn } be a maximal set of linearly independent elements of V, then {v1, v2, v3.......vn } is a basis of V.

3) Let V be a vector space consisting of n elements. Let W be a subspace which does not consist of zero alone. Then W has a basis and the dimension of W is less than or equal to n.

4) Let V be a vector space over the field F and let U,W be subspaces. If: U + W = V and if U/\W = {0} then V is the direct sum of U and W. (Note: /\ denotes intersection)

5) If V is a finite dimensional vector space over F, and is the direct sum of subspaces U, W then:

dim V = dim U + dim W

Problems:

1) Let V = R^2 and let W be the subspace (2,1). Let U be the subspace generated by (0, 1). Show that V is the direct sum of W and U.

2) Prove theorems (1) - (4)

3) what is the dimension of the space of 2 x 2 matrices? Give a basis for this space.

4) What is the dimension of the space of m x n matrices? Give a basis for this space.

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