1.3 Reduced Row-Echelon Form
Define row-equivalent matrices
one can be obtained from the other by a sequence of row operations
Define augmented matrix
suppose we have a system of m equations in n variables, with coefficient matrix A and vector of constants b; the m × (n + 1) matrix whose first n columns are the columns of A and whose last column (n + 1) is the column vector b
Define zero vector ( of size m)
the column vector of size m where each entry is the number zero
Define leading 1
the leftmost nonzero entry of a nonzero row
Theorem RREFU
(Reduced Row-Echelon Form is Unique); Suppose that A is an m × n matrix and that B and C are m × n matrices that are row-equivalent to A and in reduced row-echelon form. Then B = C.
Theorem REMES
(Row-Equivalent Matrices represent Equivalent Systems); Suppose that A and B are row-equivalent augmented matrices. Then the systems of linear equations that they represent are equivalent systems.
Theorem REMEF
(Row-Equivalent Matrix in Echelon Form); Suppose A is a matrix. Then there is a matrix B so that 1. A and B are row-equivalent 2. B is in reduced row-echelon form
2. Use row operations to convert the matrix below to reduced row-echelon form and report the final matrix.
**on loose leaf paper
3. Find all the solutions to the system below by using an augmented matrix and row operations. Report your final matrix in reduced row-echelon form and the set of solutions.
**on loose leaf paper
A matrix is in reduced row-echelon form if it meets all of the following conditions:
1. If there is a row where every entry is zero, then this row lies below any other row that contains a nonzero entry. 2. The leftmost nonzero entry of a row is equal to 1. 3. The leftmost nonzero entry of a row is the only nonzero entry in its column. 4. Consider any two different leftmost nonzero entries, one located in row i, column j and the other located in row s, column t. If s > i, then t > j.
1. Is the matrix below in reduced row-echelon form? Why or why not?
It is because in each row, the first nonzero number is a 1. The ones go diagonally downward starting from the first row. Each 1 has a 0 above it.
Define pivot column
a column containing a leading 1
Define m × n matrix
a rectangular layout of numbers from the complex numbers having m rows and n columns
Define zero row
a row of only zero entries
Define column vector (of size m)
an ordered list of m numbers, which is written in order vertically, starting at the top and proceeding to the bottom
Define row operation
will transform an m × n matrix into a different matrix of the same size; 1. Swap the location of two rows. 2. Multiply each entry of a single row by a nonzero quantity. 3. Multiply each entry of one row by some quantity, and add these values to the entries in the same columns of a second row. Leave the first row the same after this operation, but replace the second row by the new values.