Linear Algebra/Row Equivalence/Solutions
SolutionsEdit
 This exercise is recommended for all readers.
 Problem 1
Decide if the matrices are row equivalent.
 Answer
Bring each to reduced echelon form and compare.
 The first gives
 The first is this.
 These two are not row equivalent because they have different sizes.
 The first,
 Here the first is
 Problem 2
Describe the matrices in each of the classes represented in Example 2.10.
 Answer
First, the only matrix row equivalent to the matrix of all 's is itself (since row operations have no effect).
Second, the matrices that reduce to
have the form
(where , and and are not both zero).
Next, the matrices that reduce to
have the form
(where , and not both are zero).
Finally, the matrices that reduce to
are the nonsingular matrices. That's because a linear system for which this is the matrix of coefficients will have a unique solution, and that is the definition of nonsingular. (Another way to say the same thing is to say that they fall into none of the above classes.)
 Problem 3
Describe all matrices in the row equivalence class of these.
 Answer
 They have the form
 They have this form (for ).
 They have the form
 Problem 4
How many row equivalence classes are there?
 Answer
Infinitely many. For instance, in
each gives a different class.
 Problem 5
Can row equivalence classes contain differentsized matrices?
 Answer
No. Row operations do not change the size of a matrix.
 Problem 6
How big are the row equivalence classes?
 Show that the class of any zero matrix is finite.
 Do any other classes contain only finitely many members?
 Answer
 A row operation on a zero matrix has no effect. Thus each zero matrix is alone in its row equivalence class.
 No. Any nonzero entry can be rescaled.
 This exercise is recommended for all readers.
 Problem 7
Give two reduced echelon form matrices that have their leading entries in the same columns, but that are not row equivalent.
 Answer
Here are two.
 This exercise is recommended for all readers.
 Problem 8
Show that any two nonsingular matrices are row equivalent. Are any two singular matrices row equivalent?
 Answer
Any two nonsingular matrices have the same reduced echelon form, namely the matrix with all 's except for 's down the diagonal.
Two samesized singular matrices need not be row equivalent. For example, these two singular matrices are not row equivalent.
 This exercise is recommended for all readers.
 Problem 9
Describe all of the row equivalence classes containing these.
 matrices
 matrices
 matrices
 matrices
 Answer
Since there is one and only one reduced echelon form matrix in each class, we can just list the possible reduced echelon form matrices.
For that list, see the answer for Problem 1.5.
 Problem 10
 Show that a vector is a linear combination of members of the set if and only if there is a linear relationship where is not zero. (Hint. Watch out for the case.)
 Use that to simplify the proof of Lemma 2.5.
 Answer
 If there is a linear relationship where is not zero then we can subtract from both sides and divide by to get as a linear combination of the others. (Remark: if there are no other vectors in the set— if the relationship is, say, — then the statement is still true because the zero vector is by definition the sum of the empty set of vectors.) Conversely, if is a combination of the others then subtracting from both sides gives a relationship where at least one of the coefficients is nonzero; namely, the in front of .
 The first row is not a linear combination of the others for the reason given in the proof: in the equation of components from the column containing the leading entry of the first row, the only nonzero entry is the leading entry from the first row, so its coefficient must be zero. Thus, from the prior part of this exercise, the first row is in no linear relationship with the other rows. Thus, when considering whether the second row can be in a linear relationship with the other rows, we can leave the first row out. But now the argument just applied to the first row will apply to the second row. (That is, we are arguing here by induction.)
 This exercise is recommended for all readers.
 Problem 11
Finish the proof of Lemma 2.5.
 First illustrate the inductive step by showing that .
 Do the full inductive step: where , assume that for and deduce that also.
 Find the contradiction.
 Answer

In the equation

In the equation

From the prior item in this exercise we know that in the equation
 Problem 12
Finish the induction argument in Lemma 2.6.
 State the inductive hypothesis, Also state what must be shown to follow from that hypothesis.
 Check that the inductive hypothesis implies that in the relationship the coefficients are each zero.
 Finish the inductive step by arguing, as in the base case, that and are impossible.
 Answer
 The inductive step is to show that if the statement holds on rows through then it also holds on row . That is, we assume that , and , ..., and , and we will show that also holds (for in ).
 Corollary 2.3 gives the relationship between rows. Inside of those row vectors, consider the relationship between the entries in the column . Because by the induction hypothesis this is a row greater than the first , the row has a zero in entry (the matrix is in echelon form). But the row has a nonzero entry in column ; by definition of it is the leading entry in the first row of . Thus, in that column, the above relationship among rows resolves to this equation among numbers: , with . Therefore . With , a similar argument shows that . With those two, another turn gives that . That is, inside of the larger induction argument used to prove the entire lemma, here is an subargument by induction that shows for all in . (We won't write out the details since it is just like the induction done in Problem 11.)
 Note that the prior item of this exercise shows that the relationship between rows reduces to . Consider the column entries in this equation. By definition of as the column number of the leading entry of , the entries in this column of the other rows are zeros. Now if then the equation of entries from column would be , which is impossible as isn't zero as it leads its row. A symmetric argument shows that also is impossible.
 Problem 13
Why, in the proof of Theorem 2.7, do we bother to restrict to the nonzero rows? Why not just stick to the relationship that we began with, , with instead of , and argue using it that the only nonzero coefficient is , which is ?
 Answer
The zero rows could have nonzero coefficients, and so the statement would not be true.
 This exercise is recommended for all readers.
 Problem 14
Three truck drivers went into a roadside cafe. One truck driver purchased four sandwiches, a cup of coffee, and ten doughnuts for $ . Another driver purchased three sandwiches, a cup of coffee, and seven doughnuts for $ . What did the third truck driver pay for a sandwich, a cup of coffee, and a doughnut? (Trono 1991)
 Answer
We know that and that , and we'd like to know what is. Fortunately, is a linear combination of and . Calling the unknown price , we have this reduction.
The price paid is $ .
 Problem 15
The fact that Gaussian reduction disallows multiplication of a row by zero is needed for the proof of uniqueness of reduced echelon form, or else every matrix would be row equivalent to a matrix of all zeros. Where is it used?
 Answer
If multiplication of a row by zero were allowed then Lemma 2.6 would not hold. That is, where
all the rows of the second matrix can be expressed as linear combinations of the rows of the first, but the converse does not hold. The second row of the first matrix is not a linear combination of the rows of the second matrix.
 This exercise is recommended for all readers.
 Problem 16
The Linear Combination Lemma says which equations can be gotten from Gaussian reduction from a given linear system.
 Produce an equation not implied by this system.
 Can any equation be derived from an inconsistent system?
 Answer
 An easy answer is this:
 Every equation can be derived from an inconsistent system.
For instance, here is how to derive " " from
" ".
First,
 Problem 17
Extend the definition of row equivalence to linear systems. Under your definition, do equivalent systems have the same solution set? (Hoffman & Kunze 1971)
 Answer
Define linear systems to be equivalent if their augmented matrices are row equivalent. The proof that equivalent systems have the same solution set is easy.
 This exercise is recommended for all readers.
 Problem 18
In this matrix
the first and second columns add to the third.
 Show that remains true under any row operation.
 Make a conjecture.
 Prove that it holds.
 Answer
 The three possible row swaps are easy,
as are the three possible rescalings.
One of the six possible pivots is :
 The obvious conjecture is that row operations do not change linear relationships among columns.
 A casebycase proof follows the sketch given in the first item.
ReferencesEdit
 Hoffman, Kenneth; Kunze, Ray (1971), Linear Algebra (Second ed.), Prentice Hall
 Trono, Tony (compilier) (1991), University of Vermont Mathematics Department High School Prize Examinations 19581991, mimeograhed printing