4 Linear algebra

4.10 Basis and dimension examples

We’ve already seen a couple of examples, the most important being the standard basis of 𝔽n, the space of height n column vectors with entries in 𝔽. This standard basis was 𝐞1,,𝐞n where 𝐞i is the height n column vector with a 1 in position i and 0s elsewhere. The basis has size n, so dim𝔽n=n.

We can do a similar thing for the vector space of all m×n matrices over a field 𝔽. Let Eij be the m×n matrix with a 1 in position i,j and 0s elsewhere. Then the Eij, for 1im, 1jn are a basis of Mm×n(𝔽), which therefore has dimension mn.

Example 4.10.1.

The trace of a matrix is the sum of the elements of its leading diagonal. We will find a basis of the set S of 2×2 matrices with trace zero.

First note that this really is a vector space (a subspace of M2×2(𝔽)), 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 (abca). This matrix can be written

a(1001)+b(0100)+c(0010)

Call the three matrices above H,E,F so that our expression was aH+bE+cF. Since H,E,F are in S, they are a spanning sequence for S. You can check that they’re linearly independent, so they are a basis and dimS=3.

Example 4.10.2.

dimn[x]=n+1, because 1,x,,xn is a basis.

Example 4.10.3.

Let S=span(sin,cos), a subspace of the -vector space of all functions . We will find dimS.

The functions cos and sin are linearly independent by Example 4.6.4, and they span S by definition. Therefore they form a basis of S and dimS=2.