Exercises

The elements of a set might, themselves, be sets. For example, $\{1,\{2,3\}\}$ has two elements in it: $1$, and $\{2,3\}$. How many elements are there in the following sets?

1. i.

   $\{1,5,2,2\}$

2. ii.

   $\mathbb{Z}$

3. iii.

   $\{2,\mathbb{Z},ℝ,ℂ\}$

4. iv.

   $\{2,\{2,2+3\},\{2,\{1+4,2\}\},5^2\}$

For each of the following sets, determine whether F1 and F2 are satisfied. And if so, determine which of the remaining axioms F3, $\mathrm{\cdots },$ F11 are satisfied. Justify your answers.

1. i.

   The set of positive integers ${\mathbb{Z}}_{>0}:=\{1,2,3,4,\mathrm{\cdots }\}$, with the usual addition and multiplication.

2. ii.

   The set of 2 by 2 matrices ${\mathrm{M}}_2(ℝ)$ with coefficients in $ℝ$, with matrix addition and matrix multiplication.

3. iii.

   The set of complex polynomials ${𝒫}_4(ℂ)$ of degree less than or equal to 4 (see Example 2.1(v)), with the usual addition and multiplication of polynomials.

4. iv.

   (Bonus) The set of real numbers in the set $\mathbb{Q}+\mathbb{Q}\sqrt{2}:=\{a+b\sqrt{2}|a,b\in \mathbb{Q}\}$.

In the real numbers, subtraction is a binary operation: if $x,y\in ℝ$ then $x-y\in ℝ$. Prove that subtraction is not associative.

Let $F$ be a field. Prove that row-equivalence defines an equivalence relation on ${\mathrm{M}}_{n\times m}(F)$. In other words, check that

1. i.

   (“Reflexive”) Every matrix is row-equivalent to itself.

2. ii.

   (“Symmetric”) If $B$ is row-equivalent to $A$, then $A$ is row-equivalent to $B$.

3. iii.

   (“Transitive”) If $B$ is row-equivalent to $A$, and $C$ is row-equivalent to $B$, then $C$ is row-equivalent to $A$.

In your proof, label every field axiom that you use.

Find the reduced row echelon form of when

1. i.

   $F=ℝ$,

2. ii.

   $F={𝔽}_5$,

3. iii.

   $F={𝔽}_2$.

A student is asked to prove that there is only one multiplicative identity element in any field. In other words, that the multiplicative identity is unique. He writes the following:

[Student box]

Assume there are two different multiplicative identities, $1_a$ and $1_b$. Then

Contradiction. So the multiplicative identity is unique.

[End of Student box]

This solution has the right idea, but wouldn’t get full marks because he hasn’t explicitly said which field axioms he has used, and where. Fix this problem by writing a complete solution.

If $F$ is a field, and $0\ne x\in F$, then axiom F10 says there is a multiplicative inverse $y\in F$. Prove that the multiplicative inverse is unique, by assuming $y_1$ and $y_2$ both obey $x\cdot y_1=1$ and $x\cdot y_2=1$, and then use the field axioms to prove that $y_1=y_2$.

Let $F$ be a field, and $a,b\in F$. Prove that if $a\cdot b=0$ then either $a=0$ or $b=0$.

[ Hint: If you assume $a\cdot b=0$ and $a\ne 0$ and then you should try to use the field axioms to deduce from those assumptions that $b=0$. ]

Let $F$ be a field. Prove that $(-1)\cdot (-1)=1$.

Let $A,B\in {\mathrm{M}}_{n\times n}(ℝ)$, with coefficients, and denote by ${[A]}_{ij}$ the entry in the $i$th row and $j$th column of $A$. Recall that the matrix multiplication formula, for any $i,j=1,\mathrm{\cdots },n$ is:

Use the above formula to prove that matrix multiplication is associative; i.e. satisfies F6.

[ Hint: You might need to choose another subscript letter, in addition to $i,j,$ and $r$.]

Let $A,B\in {\mathrm{M}}_n(F)$ be invertible matrices. A student is asked to prove that ${(AB)}^{-1}=B^{-1}{A}^{-1}$. His proof goes as follows:

[Student box]

We need to prove that $(AB)(B^{-1}{A}^{-1})=I_n$. This can be done as follows:

Therefore, ${(AB)}^{-1}=B^{-1}{A}^{-1}$.

[End of Student box]

What has the student done wrong, and how might he get full marks?

Let $A,B,C\in {\mathrm{M}}_n(F)$. According to the definition, the matrix $A$ has inverse $B$ when both $AB=I_n$ and $BA=I_n$ are true. But maybe only one of those two equations is known to be true? To address this issue, a student is asked to prove directly that if $AB=I_n$ and $CA=I_n$ then $B=C$. His proof goes as follows:

[Student box]

If $AB=I_n$, then $B=A^{-1}$. If $CA=I_n$ then $C=A^{-1}$.
Therefore, $B=C$, since they are both equal to $A^{-1}$.

[End of Student box]

What has the student done wrong, and how might he get full marks?

A complex rational function is a function of the form $p(x)/q(x)$, where $p$ and $q$ are complex polynomials (see Example 2.1(v)), and $q$ is not the zero polynomial. The set of complex rational functions is denoted $ℂ(x)$, and will be studied in MATH215. Verify that $ℂ(x)$ a field with the usual addition and multiplication operations.