MAT223 CAP1

Definition 1.16 The pivot of a row in matrix is …

the leftmost non-zero entry in that row

Definition 1.18 A matrix is in row echelon form if …

1) All rows consisting only of zeros are at the bottom

2) The pivot of each non-zero row in the matrix is in a column to the right of the pivot of the row above it

Definition 1.19 A matrix is in Reduced row echelon form if …

1) The matrix is in the row echelon form

2) The pivot in each non-zero row is 1

3) Each pivot is the only non-zero entry in its column

Definition 1.26

Let C be the coefficient matrix for a system of linear equation in variables x1, …, xn. If the system of linear equation has at least one solution, we define the following.

1. We say that xi is a pivot variable for the system if …

2. We say that xi is a nonpivot variable of the system if …

1. the i th column of rref(C) has a pivot

2. the i th column of rref(C) does not have a pivot

Theorem 1.15 Row Equivalence

If two matrices are row equivalent, the the system of linear equation they represent as augmented matrices have the same solution set

Theorem 1.21 (Gauss-Jordan)

Every matrix A is row equivalent to a matrix in row echelon form B.

Furthermore, if B is in reduced row echelon form, then B is unique.

Lemma 1.27 Pivot Parameterization

If a system of linear equation has at least one solution, then the pivot variable of the system can be parameterized in terms of non-pivot variables. That is, if xk1, …, xkr are the non-pivot variables of the system, then for any pivot variable xi, we can find constants aij (include) R so that xpi = ai0 + ai1xk1 +… + airxkr

Theorem 1.30 Rouche Capelli

Suppose that a system of linear equations has augmented matrixA and coefficient matrixC. Then.

1. The last column of rref(A) has a pivot if and only if the system has no solutions.

2. The last column of rref(A) does not have a pivot and every column of rref(C) has a pivot if and only if the system has exactly one solution.

3. The last column of rref(A) does not have a pivot and rref(C) has a column without a pivot if and only of the system has infinitely many solution.