Contents:
Our strategy in solving linear systems, therefore, is to take an augmented matrix for a system and carry it by means of elementary row operations to an equivalent augmented matrix from which the solutions of the system are easily obtained.
In particular, we bring the augmented matrix to Row-Echelon Form: Row-Echelon Form A matrix is said to be in row-echelon form if All rows consisting entirely of zeros are at the bottom. In each row, the first non-zero entry form the left is a 1, called the leading 1.
The leading 1 in each row is to the right of all leading 1's in the rows above it. If, in addition, each leading 1 is the only non-zero entry in its column, then the matrix is in reduced row-echelon form. It can be proven that every matrix can be brought to row-echelon form and even to reduced row-echelon form by the use of elementary row operations. Many of the problems you will solve in linear algebra require that a matrix be converted into one of two forms, the Row Echelon Form ref and its stricter variant the Reduced Row Echelon Form rref.
These two forms will help you see the structure of what a matrix represents. If the matrix represents a system of linear equations, these forms allow one to write the solutions to the system. Every Computer Algebra System and most scientific or graphing calculators have commands which produce these forms for any matrix. The commands are often of the form rref A , for example.
If a matrix in echelon form satisfies the following conditions, then it is in reduced row echelon form:. A system of linear equations can be solved by reducing its augmented matrix into reduced echelon form. A matrix can be changed to its reduced row echelon form , or row reduced to its reduced row echelon form using the elementary row operations.
Switch row 1 and row 3.
All leading zeros are now below non-zero leading entries. Set row 2 to row 2 plus -1 times row 1.
And then these guys up here have to be zeroed out. So for that minus that, you get zero is equal to 5 minus 2, which is 3. Well, all of a sudden here, we've expressed our solution set as essentially the linear combination of the linear combination of three vectors. Reduced row echelon form matrices. That's minus 5 plus 6 is equal to 1.
In other words, subtract row 1 from row 2. This will eliminate the first entry of row 2. Multiply row 2 by 3 and row 3 by 2. Then the systems of linear equations that they represent are equivalent systems.
Here is a rehash of Example US as an exercise in using our new tools. The preceding example amply illustrates the definitions and theorems we have seen so far. But it still leaves two questions unanswered. Here is the answer to the first question, a definition of reduced row-echelon form. A matrix is in reduced row-echelon form if it meets all of the following conditions: If there is a row where every entry is zero, then this row lies below any other row that contains a nonzero entry.
The leftmost nonzero entry of a row is equal to 1. The leftmost nonzero entry of a row is the only nonzero entry in its column. A row of only zero entries is called a zero row and the leftmost nonzero entry of a nonzero row is a leading 1.
A column containing a leading 1 will be called a pivot column. The principal feature of reduced row-echelon form is the pattern of leading 1's guaranteed by conditions 2 and 4 , reminiscent of a flight of geese, or steps in a staircase, or water cascading down a mountain stream. There are a number of new terms and notation introduced in this definition, which should make you suspect that this is an important definition.
However, one important point to make here is that all of these terms and notation apply to a matrix. Sometimes we will employ these terms and sets for an augmented matrix, and other times it might be a coefficient matrix. So always give some thought to exactly which type of matrix you are analyzing.
Learn about the meaning of this term in Proof Technique C. The procedure given in the proof of Theorem REMEF can be more precisely described using a pseudo-code version of a computer program.
Single-letter variables, like m, n, i, j, r have the same meanings as above. So now we can put it all together.
Since the matrix in reduced-row echelon form has the same solution set, we can analyze the row-reduced version instead of the original matrix, viewing it as the augmented matrix of a different system of equations. The beauty of augmented matrices in reduced row-echelon form is that the solution sets to the systems they represent can be easily determined, as we will see in the next few examples and in the next section.