250 Lecture Index
The algorithm outlined before, which allows us to put any matrix in row echelon form (or in its unique reduced row echelon form) is extremely useful both for determining important properties of matrices, and for the practical application of solving linear systems (via the associated augmented matrix).
It is important to work through many examples to get a feel for the use of the algorithm, which is called the method of Gaussian Elimination; we will also give some names to various items encountered along the way.
Matrix Examples (work through step by step, using the algorithm as repeated below)
1. Find a leftmost nonzero entry in the matrix, and interchange two rows, if necessary, to bring that entry to the top row. For calculation purposes, it may be easiest to multiply by a scalar to make this leading entry be a one.
The column with such a leftmost nonzero entry is called the pivot column, and after interchange of rows, if necessary, to move a leading entry into the topmost position, that entry is said to be in a pivot position of the matrix. Thus step one really involves identifying the pivot column, and moving a row up, if necessary, so that its leading entry is in the pivot position.
2. Use the row with that pivot position to obtain zeros in all entries in the column below the leading entry, via scalar multiplication and row addition. The nonzero entry in the pivot position guarantees that this step can be accomplished.
3. Repeat the process, working on the submatrix obtained by ignoring the rows already completed. Work downward from top to bottom. This amounts to successively identifying the subsequent pivot columns, moving leading entries up into their subsequent pivot positions, and combining rows to get zeros in the column below that pivot position.
Note that these steps involve identifying pivot positions successively from top to bottom, with each such position somewhere to the right of its immediate predecessor
These steps will end at the bottom of the matrix, and the resulting matrix will be in row echelon form. (You do not HAVE to make the leading entries be ones to do this, but it is probably the simplest approach arithmetically.) At this point, all rows will have pivot positions identified, except possibly for some rows consisting entirely of zeros (which are at the bottom of the matrix.) It is not necessarily true that all columns will have pivot positions.
In order to put the matrix in reduced row echelon form, be sure that (via multiplication by scalars) each leading entry is one, if you did not do this already. Then, beginning with the lowest nonzero row,
1.
Use that row to obtain zeros in all entries in the column above that
pivot position.
2. Repeat the process,
working on the submatrix obtained by ignoring the rows already
completed, working upward from bottom to top.
These steps will end at the top of the matrix, and the resulting matrix will then be in reduced row echelon form. Note that in the end, every column that has a pivot position will have a single entry of one, and all its other entries zero.
When a matrix is in row echelon form, it is easy to count the number of nonzero rows (and this number is the same regardless of which row echelon form is used). Since a matrix is transformed into a unique corresponding reduced row echelon form using Gaussian elimination, the number of nonzero rows in reduced row echelon form (and hence in any row echelon form) is completely determined by the original matrix. Thus we can define a pair of numbers associated to a matrix that will turn out to be very important.
Def. The rank of an m x n matrix A is the number of nonzero rows in the reduced row echelonform of A. It is denoted rank(A). The nullity of A, denoted nullity(A), is defined to be the number of columns minus the rank. Thus, nullity(A) = n - rank(A).
Notice that the number of nonzero rows
is exactly the number of pivot positions, so the rank of a matrix is a
count of that number: rank(A) = # of pivot columns in
A.
Also, the number of columns without a pivot position is
exactly the total number of columns minus the nu mber of columns with
pivot positions, which is the nullity.
Systems of Linear Equations Examples (using the associated augmented matrix)
Notice that, if we put the coefficient matrix A associated to a linear system A x = b (not the augmented matrix) into row echelon form, we can consider solving the system regardless of what the "right hand side" is. It is clear that, for the system to have any solutions (that is, for the system to be consistent), the rows of zeros in the row echelon matrix must correspond to "zeros" in the right hand side. Notice that such equations, amounting to "0 = 0", are redundant, and thus it is only the number of nonzero rows (the rank!) that determines the number of genuine constraints in the original system.
If the system is consistent, then clearly each column which has a pivot corresponds to a variable in the solution which is determined (this is what we have referred to as a "basic" variable) and each column which does not have a pivot corresponds to a free variable in the solution. Since the number of pivots is the rank, and the number of columns without a pivot is the nullity, we have the following result.
Theorem
If
Ax = b
is the matrix form of a consistent syustem of linear equations,
then
(a) the number
of basic variables in a general solution
to the system equals the rank of A, and
(b) the number of free variables in a general solution
to the system equals the nullity of A.
Of course, this result requires us to determine whether the system is, in fact, consistent. Here is a summary of several ways of recognizing this property.
Tests for Consistency
The following conditions are equivalent.
(a) The system
Ax = b
of linear equations is consistent.
(b) The vector b is a
linear combination of the columns
of A.
(c) A row echelon form of [Ab]
has no pivot in its rightmost column.
(d) The rank of [Ab] equals the rank
of A.
To verify this result, notice that (a) and (c) are equivalent, since any rows of zeros reduced form of A correspond to zeros on the right hand side in the corresponding equations. Statement (d) is another way of stating (c), since it means that the number of pivot columns in [A b] is the same as the number of pivot columns in A. Finally, (a) and (b) are equivalent because the components of a solution x to A x = b are exactlythe coefficients of a linear combination of the columns of A that yields b.