Linear Algebra

Section III. Reduced Echelon Form 59
(a)
� �
1 0
0 0
(b)
� �
1 2
2 4
(c)
� �
1 1
1 3
2.14 How many row equivalence classes are there?
2.15 Can row equivalence classes contain different-sized matrices?
2.16 How big are the row equivalence classes?
(a) Show that the class of any zero matrix is finite.
(b) Do any other classes contain only finitely many members?
� 2.17 Give two reduced echelon form matrices that have their leading entries in the
same columns, but that are not row equivalent.
� 2.18 Show that any two n×n nonsingular matrices are row equivalent. Are any
two singular matrices row equivalent?
� 2.19 Describe all of the row equivalence classes containing these.
(a) 2 × 2matrices (b) 2 × 3matrices (c) 3 × 2 matrices
(d) 3×3 matrices
2.20 (a) Show that a vector � β0 is a linear combination of members of the set
{ � β1,... , � βn} if and only there is a linear relationship �0 =c0� β0 + ··· + cn� βn
where c0 is not zero. (Watch out for the � β0 = �0 case.)
(b) Derive Lemma 2.5.
� 2.21 Finish the proof of Lemma 2.5.
(a) First illustrate the inductive step by showing that ℓ2 = k2.
(b) Do the full inductive step: assume that ck is zero for 1 ≤ k