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.
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.
The dimension of the space of 2 x 2 matrices is 4. This means that any 2 x 2 matrix can be represented as a linear combination of 4 basis matrices.
A basis for the space of 2 x 2 matrices is given by:
Basis = {[
[1, 0],
[0, 0]
],
[
[0, 1],
[0, 0]
],
[
[0, 0],
[1, 0]
],
[
[0, 0],
[0, 1]
]}
Now, let's answer the same question for the space of n x m matrices.
The dimension of the space of n x m matrices is nm. This means that any n x m matrix can be represented as a linear combination of nm basis matrices.
A basis for the space of n x m matrices can be constructed by considering each element of the matrix as an independent variable. Each basis matrix will have only one non-zero element and all other elements will be zeros.
For example, for a 3 x 2 matrix, a possible basis could be:
Basis = {[
[1, 0],
[0, 0],
[0, 0]
],
[
[0, 1],
[0, 0],
[0, 0]
],
[
[0, 0],
[1, 0],
[0, 0]
],
[
[0, 0],
[0, 1],
[0, 0]
],
[
[0, 0],
[0, 0],
[1, 0]
],
[
[0, 0],
[0, 0],
[0, 1]
]}
In general, the basis will consist of nm basis matrices, each with only a single non-zero element at different positions.