Prove that the space of all \(m \times n\) matrices over the field \(F\) has dimension \(mn\), by exhibiting a basis for this space.
Solution
1. Let \(E_{ij}\) be matrix with \(1\) in \((i,j)\) and \(0\) elsewhere.
definition.
2. \(\{E_{ij}:1\le i\le m,1\le j\le n\}\) spans.
any matrix \(A=(a_{ij})=\sum a_{ij}E_{ij}\).
3. They are linearly independent: \(\sum c_{ij}E_{ij}=0\) implies \(c_{ij}=0\).
entrywise.
4. Hence basis, dimension \(mn\).
2 and 3.
5. Q.E.D.
4.