K. Nicole Clark k-clark

URL: people.tamu.edu/~knc8928/5.4.3

**5.4.3 – **(a) Find all the solutions of the
system { x + 2y – 3z = 2,

{ x – 2y + 4z = 1.

(b) Find the basis for the range of the linear function whose matrix is

_{}

(c) Comment on the relation between the number of parameters in the solution to (a) and the number of vectors in the solution to (b).

**Solution**

(a) This is a simple Gauss –

_{}_{}

_{}

(b) The range of a function is the span of its columns. We can thus find the transpose of the matrix and then row reduce it to find the basis.

_{}

Thus, a basis for this matrix is _{}.

(c) The dimension of the range
plus the dimension of the kernel must equal the dimension of the domain. In this case, the dimension of the range,
otherwise known as the *rank*, is 2,
the dimension of the domain is 3, thus it follows that the dimension of our
solution space should be 1, as it is equal to the dimension of the kernel.