Math Revisited: Linear Algebra (Dimension)

The concept of 'dimension' in linear algebra basically has direct applicability for all finite dimensional vector spaces, up to some limit, n. There also occur infinite dimensional vector spaces for which it's possible to provide an infinite basis, but these are more peculiar to specialized applications as in advanced quantum theory (e.g. Hilbert spaces as infinite dimensional vector spaces). We will eschew this special version until a much later time, next year, when we survey some topics in advanced quantum mechanics.

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 Rⁿ     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  Fⁿ  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 Auxiliary 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 = R² 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.