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 space of 2x2 matrices has dimension 4. A basis for this space can be given by the following matrices:
1) {{1, 0}, {0, 0}}
2) {{0, 1}, {0, 0}}
3) {{0, 0}, {1, 0}}
4) {{0, 0}, {0, 1}}
For the space of n x m matrices, the dimension is given by n*m. A basis for this space can be constructed by creating a matrix with all zeroes except for a single 1 in each entry. For example, for a 3 x 2 matrix, a possible basis would be:
1) {{1, 0}, {0, 0}, {0, 0}}
2) {{0, 1}, {0, 0}, {0, 0}}
3) {{0, 0}, {1, 0}, {0, 0}}
4) {{0, 0}, {0, 1}, {0, 0}}
5) {{0, 0}, {0, 0}, {1, 0}}
6) {{0, 0}, {0, 0}, {0, 1}}
The dimension of the space of 2x2 matrices is 4. To find a basis for this space, we can consider the matrices with only one non-zero entry.
A basis for the space of 2x2 matrices can be {[1, 0; 0, 0], [0, 1; 0, 0], [0, 0; 1, 0], [0, 0; 0, 1]}.
For the space of n x m matrices, the dimension is given by the product of n and m. To find a basis for this space, we can consider the matrices with only one non-zero entry.
A basis for the space of n x m matrices can be obtained by considering the matrices with a single non-zero entry of value 1, located at different positions.