Solved–Homework 3– Solution

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1. We discussed in class the concept of equivalence relations and then studied a very speci c equivalence relation de ned over the set M_{m n}(R). The goal of this question is to review these concepts.

1. 1. Denote by S the set of all students in our class. Give examples of three di erent relations that one can de ne over the set S which are equivalence relations. Explain why you claim that they are equivalence relations. For example, you can check that the following relation is an

equivalence relation: “For a; b 2 S we say that a b if b has the same amount of siblings as a does”.

1. 2. Denote by S the set of all students in our class. Give examples of three di erent relations that one can de ne over the set S which are not (necessarily) equivalence relations. Explain why you claim that they

are not (necessarily) equivalence relations. For example: the relation

de ned by: “For a; b 2 S we say that a b if b enjoys the company of a“, is not (necessarily) an equivalence relation as (unfortunately) it may happen that this relation is not symmetric (that is, b might enjoy the company of a while a does not enjoy the company of b so much).

It also may happen that this relation is not transitive (that is, it can happen that c enjoys the company of b, and b enjoys that of a, however

c does not enjoy being around a at all). In fact, this relation might even

not be re exive (it can happen that a person extremely su ers from the company of himself…).

3. The relation de ned over R² by (x₁; y₁) (x₂; y₂) if x₁ = x_{2 is an} equivalence relation. Prove this fact. Can you describe the equivalence classes obtained by this equivalence relation?

3. The relation de ned over R² by (x₁; y₁) (x₂; y₂) if x₁ + x₂ = 0 is not an equivalence relation. Prove this fact.

3. In class we de ned a very speci c relation over M_{m n}(R) and proved that this speci c relation is an equivalence relation. Complete this sen-tence: “For A; B 2 M_{m n}(R) we de ned that A B if …”. Recall that in this case, we say that A and B are row equivalent.

3. For the speci c equivalence relation de ned in (v), give examples of 3

di erent matrices which are in the equivalence class of the matrix

0 1

1	1	1	A :
@ 0	0	3

• 2 8

(That is, give examples of matrices which are row equivalent to this matrix).

2

2. In each of the following parts determine whether the two given matrices are row equivalent. (We discussed how such a question can be solved in class.

Examples will be given in the next recitation.)

i.	(	3		1	) and (				4	8	)
	(	1		3					1	2			)
ii.			1	2	2	) and (				2	2	5
	0	1		1	1		1			0	3	1		1
iii.			0		0	3		and		0	1	0	3
	@		1		1	1	A			@	0	1	2	A
			2		2	8					1	1	1

3. Prove the following statements.

   1. Let A 2 M_{m n}(R). If an echelon form of A has a row of zeroes then there exists b 2 R^m such that (Ajb) has no solution.

3. 2. If m > n and A 2 M_{m n}(R) then there exists b 2 R^m such that (Ajb) has no solution.

3. 3. If A 2 M_n(R) is a square n n matrix such that the homogenous system (Aj0) has in nity many solutions, then there exists b 2 R^{n such that} (Ajb) has no solution.

3. Let A 2 M_n(R) be a square n n matrix. Prove that the following two statements are equivalent, that is, prove that a ) b and b ) a (one can also say that “a is true if and only if b is true”).

3. 1. The homogeneous system (Aj0) has exactly one solution.

3. 2. For every b 2 Rⁿ the linear system (Ajb) has a solution.

3. The following statements are false. Prove that they are false by providing a counterexample in each case (you may choose the numbers m and n to be whatever is convenient, try to work with small numbers).

3. 1. If A 2 M_{m n}(R) is a matrix such that the homogenous system (Aj0) has in nity many solutions, then there exists b 2 Rⁿ such that (Ajb) has no solution. (Note that by question (4), we know that this statement is in fact true if A is a square matrix).

3. 2. Let A; B 2 M_{m n}(R) and b 2 R^m. If A and B are row equivalent then (Ajb) and (Bjb) have the same amount of solutions.

3. 3. If A; B 2 M_{m n}(R) are row equivalent then B can be obtained from A by performing column operations (that is, by performing a sequence of operations of the form ‘swapping two columns’, ‘multiplying a column by a scalar di erent from zero’, ‘adding to a column another column multiplied by a scalar’).

3

4. Let A 2 M_{m n}(R) and b 2 R^m. If u; v 2 Sol(Ajb) then u + v 2 Sol(Ajb).

4. Let A 2 M_{m n}(R) and b 2 R^m. If u 2 Sol(Ajb) and t 2 R then tu 2 Sol(Ajb).