III. Reduced Echelon Form Example: Echelon forms are not unique. III.1. Gauss-Jordan Reduction III.2. Row Equivalence

III.1. Gauss-Jordan Reduction Definition 1.3: A matrix is in reduced echelon form if, in addition to being in echelon form, each leading entry is a one and is the only nonzero entry in its column. Example 1.1: Gauss-Jordan method: Solution via reduced echelon form.

Example ( reduced echelon form )

Properties of the reduced echelon form: It is equivalent to parametrization by unmodified free variables. It is unique (to be proved later). Matrices sharing the same reduced echelon form are “equivalent”. Definition: Equivalence Relation ~ is an equivalence relation if it is Reflexive: a ~ a. Symmetric: a ~ b → b ~ a. Transitive: a ~ b & b ~ c → a ~ c.

Lemma 1.4: Elementary row operations are reversible. Proof: Row swapping (ρi  ρj ) : Inverse = ρj  ρi . Multiplication by constant (ρi → k ρi ) : Inverse = ρi → k1 ρi Adding k  row i to row j (ρj → k ρi + ρj ) : Inverse = ρj →  k ρi + ρj Lemma 1.5: Between matrices, ‘reduces to’ is an equivalence relation. Proof: 1. Reflexivity: Any matrix reduces to itself through no row operations. 2. Symmetry: Follows from lemma 1.4. 3. Transitivity: A reduction from A to B followed by that from B to C gives a reduction from A to C.

Definition 1.6: Row Equivalence 2 matrices related by elementary row operations (interreducible) are row equivalent. Row equivalence partitions the collection of matrices into row equivalence classes. Since the reduced echelon form is unique, we have 1. Every matrix is row-equivalent to a unique reduced echelon form. 2. Every row equivalence class contains 1 and only 1 reduced echelon form matrix. 3. A row equivalence class can be represented by its reduced echelon form matrix. 4. Two matrices are in the same row equivalence class iff they share the same reduced echelon form.

Exercises 1.III.1 1. List the reduced echelon forms possible for each size. (a) 22 (b) 2  3 (c) 3  2 (d) 3  3 Hint: Answer for (a) is given in Example 2.10.

III.2. Row Equivalence Definition 2.1: Linear Combination A linear combination of x1 , …, xm is an expression of the form c1 x1 + …+ cm xm where ci are scalars Lemma 2.2: Linear Combination Lemma A linear combination of linear combinations is a linear combination. Proof: By collecting coefficients. Notation: The ith row of matrix A is represented by αi . Corollary 2.3: Where one matrix row reduces to another, each row of the second is a linear combination of the rows of the first. Proof: By induction.

Lemma 2.5: In an echelon form matrix, no nonzero row is a linear combination of the other rows. I.e., all non-zero rows are linearly independent. Proof: By contradiction. Theorem 2.7: Each matrix is row equivalent to a unique reduced echelon form matrix. Proof: By construction via Gauss-Jordan reduction. Example 2.8: → A and B are not row equivalent.

Example 2.9: Any nn non-singular matrix is row equivalent to the nn unit matrix. Example 2.10: The possible row equivalent classes for 22 matrices are

Exercises 1.III.2 Give two reduced echelon form matrices that have their leading entries in the same columns, but that are not row equivalent. In this matrix the first and second columns add to the third. (a) Show that remains true under any row operation. (b) Make a conjecture. (c) Prove that it holds.