just need to know how to solve a rank and null space of an 3x5 matrix. any help would be helpful thanks.
The rank of a matrix is the dimension of its row or column space (both will be the same). Now, you could find a basis for the row space, and then the number of elements in the basis will be the rank of the matrix. To find a basis for the row space, you can use the fact that the nonzero rows of a matrix in row-echelon that is row-equivalent to another matrix A will actually form a basis for the row space of A.
So, for example:
Put A in row echelon form to get:
Now, since has two nonzero rows, you can conclude that .
However, once you know the nullspace of a matrix, you usually don't need to go through the above process to find the rank.
To find the nullspace of A, simply solve the system .
For example, using the same matrix:
Augment this matrix with the 0 column vector and reduce:
So,
The dimension of the nullspace is called the nullity. Knowing the nullity of a matrix allows you to find the rank very easily, without finding the row or column space: For any matrix, .
For our matrix, we see that , which agrees with what we found earlier.
I hope that helped!