What is the dimension of the space of 2 x 2 matrices? Give a basis for this space. Answer the same question for the space of n x m matrices.

Answer 1

The dimension of the space of 2 x 2 matrices is 4, since each matrix has 4 entries.

A basis for this space can be {E11, E12, E21, E22}, where Eij denotes the matrix with 1 in the (i, j) entry and 0 elsewhere. In other words:

E11 = |1 0|
|0 0|

E12 = |0 1|
|0 0|

E21 = |0 0|
|1 0|

E22 = |0 0|
|0 1|

Answering the same question for the space of n x m matrices, the dimension of this space is nm, since each matrix has nm entries.

A basis for this space can be {Eij}, where Eij denotes the matrix with 1 in the (i, j) entry and 0 elsewhere. This basis consists of nm matrices in total.