Final answer:
To find the basis and dimension of the space of all 2x2 matrices with elements from the set of real numbers, we need to set up a system of equations with the matrix entries as variables. Solving this system will give us the basis and dimension of the space.
Step-by-step explanation:
To find the basis and determine the dimension of the space of all 2x2 matrices with elements from the set of real numbers, we first need to determine the span of this set of matrices. Let's denote the set of matrices as M:
M = {{mat1}, {mat2}, ...}
Then, the basis for M will be a set of linearly independent matrices that span M. In this case, the dimension of M will be the number of linearly independent matrices in the basis set.
To find the basis and dimension, we can start by considering the entries of the matrices as variables and set up a system of equations:
3a - c = 0
-b - 3c = 0
-7a + 6b + 5c = 0
-3a + c = 0
Solving this system of equations, we can find the values of a, b, and c that satisfy the equations. From there, we can determine the basis and dimension of M.