Home page for accesible maths 7 Coursework 7 Coursework 7.B Week 2

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7.A Week 1

In-class exercises

1.2, 1.14, 1.15, 1.16, 1.17(i), 1.20, 2.2, 2.4, 2.11(i)(ii)

Unassessed exercises

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Top priority: 1.11, 1.17(iii), 1.22, 1.26, 2.6, 2.11(iv)(v)
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Other exercises: 1.18, 1.19, 1.23, 1.27, 1.28, 2.12(i)(ii)

Written assignment (Due date: 2:00pm, Thursday 12 October 2017)

[30 marks total. Please staple your pages together, and acknowledge sources of help.]

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1.8 [1 mark]
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1.17(ii) [8 marks]
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1.21 [3 marks]
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1.25 [5 marks]
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2.10 [3 marks]
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2.11(iii) [2 marks]
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2.54 [4 marks]

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Reflection A [2 marks]: Try coming up with a different word or name, instead of “field”, for a concept that includes ${\mathbb{Q}},{\mathbb{R}}$ , and ${\mathbb{C}}$ . Discuss why you think your name is better or worse than the word “field”. A few sentences will suffice. You will receive full marks for this question if you can convince the marker that you have given some thought to your choice.
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Reflection B [2 marks]: When you consider a polynomial, such as $x^{2}+3x+2$ , what goes through your mind? Do you think of it as a function, where $x$ is a variable; do you think of it as a number, where $x$ is some yet-to-be-determined value; or perhaps you think of $x$ as an abstract symbol without any particular meaning? Try to give some reasons why you conceptualize polynomials in the way that you do. You will receive full marks for this question if you can convince the marker that you have given some thought to your answer.

Weekly true / false quiz (closes at 2:00pm, Saturday 14 October 2017)

[10 marks total.]

Q1.

– Fields

A field is a set together with an addition and multiplication operation that obeys the list of axioms given in the notes. Determine which of the following statements are true.
- (a)
  
  If $F$ is a field, then its multiplication is always associative,
- (b)
  
  If $F$ is a set with an addition and multiplication which are both associative, then $F$ is a field,
- (c)
  
  If $F={\mathbb{Q}}$ then $F$ is a field,
- (d)
  
  If $F$ is a field then either $F={\mathbb{Q}}$ , $F={\mathbb{R}}$ , or $F={\mathbb{C}}$ .
Q2.

– Elements of sets.
The symbol $\forall$ means “for all”; $\exists$ means “exists”; $\in$ means “element of”; and $\notin$ means “not an element of”. Recall that we have inclusions of fields ${\mathbb{Q}}\subset{\mathbb{R}}\subset{\mathbb{C}}$ . Determine which of the following statements are true:
- (a)
  
  $\exists x\in{\mathbb{R}}$ such that $x\notin{\mathbb{C}}$ ,
- (b)
  
  $\forall x\in{\mathbb{C}}$ , we have $x\neq 0$ ,
- (c)
  
  $\exists x\in{\mathbb{C}}$ such that $x\neq 0$ ,
- (d)
  
  Let $A\subset{\mathbb{C}}$ be a subset. Assume that $\forall x\in A$ we have $x\in{\mathbb{Q}}$ . This assumption implies that $A\subset{\mathbb{R}}$ .
Q3.

– Complex conjugate
The complex conjugate of a complex number is defined to be: $\overline{a+bi}:=a-bi$ , where $a,b\in{\mathbb{R}}$ . Determine which of the following statements are true:
- (a)
  
  If $x=a+bi\in{\mathbb{C}}$ then $x\overline{x}=a^{2}-b^{2}$ ,
- (b)
  
  $\forall x\in{\mathbb{C}}$ , the number $x\overline{x}$ is a real number,
- (c)
  
  $\exists x\in{\mathbb{C}}$ , such that the number $x\overline{x}$ is a negative real number,
- (d)
  
  The following sets are equal: $\{x\in{\mathbb{C}}\;|\;x=\overline{x}\}={\mathbb{R}}$ .
Q4.

– F24
Consider the set $\mathbb{F}_{24}:=\{0,1,2,\cdots,22,23\}$ of integers modulo 24, with the addition and multiplication analogous to that used in Exercise 1.2. This is like the hours in a 24 hour clock. For example, 22:00 + 7 hours = 05:00, so in $\mathbb{F}_{24}$ we have $22+7=5$ . One can also verify that $5\cdot 5=1$ and $7\cdot 7=1$ . Determine which of the following statements are true:
- (a)
  
  In ${\mathbb{F}}_{24}$ we have $5\cdot 7=10$ ,
- (b)
  
  In ${\mathbb{F}}_{24}$ , we have $20^{2}=4^{2}$ ,
- (c)
  
  In ${\mathbb{F}}_{24}$ , we have $(18+10)\cdot(7+5)\neq 0$ ,
- (d)
  
  In ${\mathbb{F}}_{24}$ , the axiom F10 is satisfied.
Q5.

– Upper triangular

Assume that $A,B\in\operatorname{M}_{{n}}({{\mathbb{R}}})$ are two upper triangular matrices. Recall that a matrix is upper triangular if all entries below the diagonal are zero. A student is asked to prove that $A B$ must also be an upper triangular matrix. She writes the following:

[Student box]
- (1)
  
  Let $A=[a_{ij}]$ , $B=[b_{ij}]$ . If $i>j$ then $a_{ij}=b_{ij}=0$ , since $A$ and $B$ are upper triangular, .
- (2)
  
  The matrix multiplication formula is: $[AB]_{ij}=\sum_{r=1}^{n}a_{ir}b_{rj}$ .
- (3)
  
  If $i>j$ , and $r$ is any integer such that $1\leq r\leq n$ , then either $i>r$ or $r>j$ (or both).
- (4)
  
  Combining Line (1) and (3), for any $r=1,\cdots,n$ we know $a_{ir}\cdot b_{rj}=0$ .
- (5)
  
  Therefore, if $i>j$ then $[AB]_{ij}=\sum_{r=1}^{j}0=0$ , and so $A B$ is upper triangular.
[End of Student box]
Determine which of the following statements are true:
- (a)
  
  The conclusion in Line 1 is incorrect; it should instead say $i<j$ ,
- (b)
  
  The statement in Line 3 is still true if we let $r$ be any integer,
- (c)
  
  Line 4 used that if $x$ or $y$ equals zero then $x\cdot y=0$ ,
- (d)
  
  The formula in Line 5 correctly combines Lines 2 and 4.