If M is an m \times n matrix and f is the function from R^{n} to R^{m} that is given by f(x) = M x for x in R^{n}, then:
The image of f is a linear subspace of R^{m}.
The kernel of f is a linear subspace of R^{n}.
The rank of f is the dimension of the image of f. It is equal to the rank of M.
The nullity of f is the dimension of the kernel of f.
One has the formula
Note: n is the dimension of the domain of f. It is not necessarily equal to m, which is the dimension of the target of f.
Let f: R^{4} -----> R^{3} be the map defined by f(x) = M x, where M is the matrix
| -2 | 1 | -3 | 2 |
| 1 | 0 | 1 | -1 |
| 3 | -1 | -3 | 4 |
) .
What is the rank of f ?
What is the nullity of f ?
Find a parametric representation of the fiber of f over the point (-4, 2, -1).
Find a parametric representation of the fiber of f over the point (0, 0, 7).
Find a parametric representation of the fiber of f over the point (1, -2, 3).