Linear Algebra

Section III. Reduced Echelon Form 51
(a) 2x + y − z =1
4x − y =3
(b) x − z =1
y +2z − w =3
x +2y +3z − w =7
(d) a +2b +3c + d − e =1
3a − b + c + d + e =3
1.10 Give two distinct echelon form versions of this matrix.
� �
2 1 1 3
6 4 1 2
1 5 1 5
(c) x − y + z =0
y + w =0
3x − 2y +3z + w =0
−y − w =0
� 1.11 List the reduced echelon forms possible for each size.
(a) 2×2 (b) 2×3 (c) 3×2 (d) 3×3
� 1.12 What results from applying Gauss-Jordan reduction to a nonsingular matrix?
1.13 The proof of Lemma 1.4 contains a reference to the i �= j condition on the
row pivoting operation.
(a) The definition of row operations has an i �= j condition on the swap operation
ρi ↔ ρj. Show that in A ρi↔ρj −→ ρi↔ρj −→ A this condition is not needed.
(b) Write down a 2×2 matrix with nonzero entries, and show that the −1·ρ1 +ρ1
operation is not reversed by 1 · ρ1 + ρ1.
(c) Expand the proof of that lemma to make explicit exactly where the i �= j
condition on pivoting is used.
1.III.2 Row Equivalence
We will close this section and this chapter by proving that every matrix is
row equivalent to one and only one reduced echelon form matrix. The ideas
that appear here will reappear, and be further developed, in the next chapter.
The underlying theme here is that one way to understand a mathematical
situation is by being able to classify the cases that can happen. We have met this
theme several times already. We have classified solution sets of linear systems
into the no-elements, one-element, and infinitely-many elements cases. We have
also classified linear systems with the same number of equations as unknowns
into the nonsingular and singular cases. We adopted these classifications because
they give us a way to understand the situations that we were investigating. Here,
where we are investigating row equivalence, we know that the set of all matrices
breaks into the row equivalence classes. When we finish the proof here, we will
have a way to understand each of those classes — its matrices can be thought
of as derived by row operations from the unique reduced echelon form matrix
in that class.
To understand how row operations act to transform one matrix into another,
we consider the effect that they have on the parts of a matrix. The crucial
observation is that row operations combine the rows linearly.