K. Nicole Clark                                                                                                           k-clark

3/22/2004                                                                                                         Math 311-200

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 – Jordan elimination problem.  We start by forming an augmented matrix, and then row reduce to find the values for x, y, and z.

(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.