We’ve already seen a couple of examples, the most important being the standard basis of , the space of height n column vectors with entries in . This standard basis was where is the height n column vector with a 1 in position i and 0s elsewhere. The basis has size n, so .
We can do a similar thing for the vector space of all matrices over a field . Let be the matrix with a 1 in position and 0s elsewhere. Then the , for , are a basis of , which therefore has dimension .
The trace of a matrix is the sum of the elements of its leading diagonal. We will find a basis of the set of matrices with trace zero.
First note that this really is a vector space (a subspace of ), so its dimension is at most 4.
A good start is to write down an expression for a general matrix with trace zero. It must have the form . This matrix can be written
Call the three matrices above so that our expression was . Since are in , they are a spanning sequence for . You can check that they’re linearly independent, so they are a basis and .
, because is a basis.
Let , a subspace of the -vector space of all functions . We will find .
The functions and are linearly independent by Example 4.6.4, and they span by definition. Therefore they form a basis of and .