北京邮电大学国际学院线性代数讲义Lecture 02汇总

Using row operation III to eliminate the nonzero entries in the last four rows of the first column, the resulting matrix will be
Row Echelon Form
Example: (continue)
1 0 0 0 0
1 0 0 0 0
1 1 2 1 1
1 1 2 1 1
1 1 2 0 Leabharlann Baidu pivotal row 5 3 3 1 3 0
Row Echelon Form
Example: (continue) Continue the eliminating process, we have
Will be satisfied for any 5-tuple.
Row Echelon Form
and the solution set will be the solution set of all 5-tuples satisfying the first three equations
x1
1 0 0 1 2 1 1 0 0 0
1 0 0 1 2 1 1 0 0 0
1 1 1 1 3 1 3 3 4 4 1 2 1 0 0 1 0 3 0 0
Then, the reduction process will yield the augmented matrix
Row Echelon Form
If we change the right hand of equations in last example, so as to obtain a system has solution set, such as
1 1 1 1 2 2 0 0 1 1 1 0 0 0 0 1 0 0 0 0
Row Echelon Form
Example: Consider the system represented by the augmented matrix
1 1 1 1 2 2 0 0 1 1
1 0 0 1 2
1 0 0 1 2
1 1 pivotal row 1 1 3 1 3 1 4 1
1 0 0 0 0
and we end up with
1 0 0 0 0
1 1 0 0 0 1 0 0 0 0
1 1 0 0 0 1 0 0 0 0
1 1 2 0 1 3 pivotal row 1 1 1 0 1 1 0 0 0 1 1 0 0 0 1 1 2 0 1 3. 0 4 0 3
1 0 0 0 0
1 0 0 0 0
1 1 2 1 1
1 1 2 1 1
1 1 2 0 pivotal row 5 3 3 1 3 0
It is easy to be seen that all possible choices of pivot are 0 in the second column. It is nature to continue our reduction process from the next column.
Row Echelon Form
Example: (continue) By augmented matrix
1 0 0 0 0
The last two rows mean that 0 x1 0 x2 0 x1 0 x2 has no solution.
1 0 0 0 0
Lecture 2
Row Echelon Form
Row Echelon Form
In last Lecture, we had seen that a n n system may be reduced to triangular form. But this progress may be broken if at any step, all the possible choices for a pivot element are 0. Since, the progress of reducing a system to triangular form is a progress of eliminating variables, at the stage of the reduction breaks down, it seems natural to move over the next column and eliminate the rest variables. By doing this, it is clear that the equations system can not be reduced to triangular form.

x2

x3 x3

x4 x4
x5 2 x5 x5
1 0 3
Definition: The variables corresponding to the first nonzero elements in each row of the augmented matrix will be referred to as the lead variables and the remaining variables corresponding to the columns skipped in the reduction process will be referred to as free variables.
1 1 0 0 0
1 1 0 0 0
1 1 2 0 1 3 0 4 0 3
0 x5 0 x5 4 3
0 x3 0 x3
0 x4 0 x4
Since, there are no 5-tuples that could possibly satisfy these equations, the system