VCE Specialist Mathematics Units 3&4: Concise Guides

[AlphaZero]

VCE Specialist Mathematics Units 3&4: Concise Guides
by AlphaZero (Dan)

How will this work?
Basically, you folks request a topic that you would like me to write a concise guide on! This can be any topic you like. For example: implicit differentiation, sketching rational functions, hypothesis testing, volumes of revolution, etc. I going to aim to produce around one guide every 1-2 weeks, but keep in mind, I'm a very busy person, so I apologise in advance if you don't see anything from me for some time. If I get lots of different requests, I'll pick the most popular.

But...
Please remember that these are meant to be miniature guides for people to learn from. Do not message me with "can you write a guide on calculus?" There are entire textbooks on calculus. To write a 'concise' guide on such a topic would be futile due the amount of detail I would need to omit to make it even remotely brief. And, as most of you would know, omitting details, particularly in this subject, does not serve students well.

Contents
1.  Linear Dependence and Independence

To request a topic, reply to this thread or send me a message!

[S_R_K]

A few topics that it might be useful to have guides on because the popular textbooks contain unsatisfactory presentations:

1. Rational functions, including how to sketch efficiently without CAS, and how to identify key features of families of rational functions.

2. Properties of composite functions involving the inverse circular functions (ie. things like finding the maximal domain and range of arcsin(sqrt(x^2 – 1)))

3. Graphs of complex number relations that are in less familiar forms (ie. az + bz* = c).

[AlphaZero]

1.  Linear Dependence and Independence
01 August 2019

Prerequisite Knowledge
> A basic understanding of vectors
Introduction

The concept of linear dependence/independence is one that isn't really well investigated in my opinion. VCE students upon encountering this concept may question why they are bothering to learn this, when it seems so abstract and perhaps a bit random. Fear not - those who will pursue studies in mathematics will encounter the world of linear algebra, where this becomes very important. In this guide, I'll hopefully clear up some misconceptions and/or muddy points about linear dependence/independence!
Definitions

All three of the following definitions regarding linear dependence/independence are equivalent.

Definition 1: A set of vectors \(\{\mathbf{v}_1,\mathbf{v}_2,\dots,\mathbf{v}_n\}\) is said to be linearly dependent if there exists \(k_1,\dots,k_n\in\mathbb{R}\), not all zero, such that \(k_1\mathbf{v}_1+\dots+k_n\mathbf{v}_n=\mathbf{0}\).

Definition 2: A set of vectors is said to be linearly dependent if at least one of its members can be expressed as a linear combination of the remaining members.

Definition 3: A set of vectors \(\{\mathbf{v}_1,\mathbf{v}_2,\dots,\mathbf{v}_n\}\) is said to be linearly independent if \(k_1\mathbf{v}_1+\dots+k_n\mathbf{v}_n=\mathbf{0}\implies k_1=\dots=k_n=0\).

Essentially, to determine whether a set of vectors is linearly dependent or independent, you should form the linear system given by \[k_1\mathbf{v}_1+\dots+k_n\mathbf{v}_n=\mathbf{0}\] and aim to solve for \(k_1,\dots,k_n\). Using our definitions from above, if you find that the only solution is \(k_1=\dots=k_n=0\) (note that this will always be a solution), then the set is linearly independent. If you find there are alternative solutions for which not all \(k_i\) are \(0\), then the set is linearly dependent.
A Few Useful Theorems

Theorem 1: Two vectors are linearly dependent if and only if they are proportional.

You should be able to prove this yourself

\((\implies)\)  Suppose that \(\mathbf{a}\) and \(\mathbf{b}\) are linearly dependent. Since there are only two vectors, by Definition 2, we can write \(\mathbf{a}\) as a linear 'combination' of the remaining vector \(\mathbf{b}\). That is, for some \(k\in\mathbb{R}\), we have \(\mathbf{a}=k\mathbf{b}\), and so \(\mathbf{a}\) and \(\mathbf{b}\) are proportional.

\((\impliedby)\)  Suppose that \(\mathbf{a}\) and \(\mathbf{b}\) are proportional. That is, for some \(k\in\mathbb{R}\), we have \(\mathbf{a}=k\mathbf{b}\). Then, \(\mathbf{a}-k\mathbf{b}=\mathbf{0}\), and so by Definition 1, \(\mathbf{a}\) and \(\mathbf{b}\) are linearly dependent.  \(\Box\)

Theorem 2: A set of \(n+1\) vectors in \(\mathbb{R}^n\) is necessarily linearly dependent.
(You do not need to be able to prove this).
Some Common Misconceptions

Misconception 1: If the vectors \(\mathbf{a},\mathbf{b},\mathbf{c}\in\mathbb{R}^3\) are linearly dependent then \(\mathbf{c}=k_1\mathbf{a}+k_2\mathbf{b}\) for some \(k_1,k_2\in\mathbb{R}\).

Misconception 2: The vectors \(\mathbf{a},\mathbf{b},\mathbf{c}\in\mathbb{R}^3\) are linearly independent if \(\mathbf{c}\neq k_1\mathbf{a}+k_2\mathbf{b}\) for all \(k_1,k_2\in\mathbb{R}\).

These are both incorrect. Definition 2 states that a set of vectors is linearly dependent if at least one of its members can be expressed as a linear combination of the remaining members. That is, \(\mathbf{c}\) need not be that vector. Consider the vectors \[\mathbf{a}=2\mathbf{i}-\mathbf{j}+\mathbf{k},\ \ \mathbf{b}=-4\mathbf{i}+2\mathbf{j}-2\mathbf{k},\ \ \mathbf{c}=3\mathbf{i}+2\mathbf{j}+\mathbf{k}.\] It can very easily be shown that  \(\mathbf{c}\neq m\mathbf{a}+n\mathbf{b}\ \ \forall\,m,n\in\mathbb{R}\)  yet its obvious that  \(\mathbf{a},\mathbf{b},\mathbf{c}\)  are linearly dependent since \[2\mathbf{a}+\mathbf{b}+0\mathbf{c}=\mathbf{0}.\] Here, either \(\mathbf{a}\) or \(\mathbf{b}\) can be written as a linear combination of \(\mathbf{b}\) and \(\mathbf{a}\) respectively.

The converse of the above statements are true.
Example (from TWM Publications Free Specialist Exam 2)

Question 14
The vectors  \(\mathbf{a}=3\mathbf{i}-3\mathbf{j}+\alpha\mathbf{k}\),  \(\mathbf{b}=\mathbf{i}+\mathbf{j}-3\mathbf{k}\)  and  \(\mathbf{c}=2\mathbf{i}-\mathbf{j}+4\mathbf{k}\)  are linearly independent if
A.    \(\alpha =1\)
B.    \(\alpha =11\)
C.    \(\alpha \in \mathbb{R}\setminus \{1\}\)
D.    \(\alpha \in \mathbb{R}\setminus \{11\}\)
E.    \(\alpha \in \mathbb{R}\)

Solution

Since \(\mathbf{b}\) and \(\mathbf{c}\) are not proportional, we consider \(\mathbf{a}=k_1\mathbf{b}+k_2\mathbf{c}\) and aim for linear dependence. Comparing vector components, we have \[\begin{cases} 3=k_1+2k_2\\ -3=k_1-k_2\\ \alpha=-3k_1+4k_2\end{cases}\]Solving the linear system gives \(\alpha=11,\ k_1=-1,\ k_2=2\), so that \(\mathbf{a},\mathbf{b},\mathbf{c}\) are linearly dependent. Thus, for linear independence, we require \(\alpha\neq 11\).

The answer is D.

A Useful Test for Linear Independence/Dependence (for Exam 2)

Let \(\mathbf{v}_1,...,\mathbf{v}_n\in\mathbb{R}^n\) and construct the matrix \[A=\begin{bmatrix}\mid &  & \mid \\ \mathbf{v}_1 & \dots & \mathbf{v}_n\\ \mid & & \mid \end{bmatrix}\] obtained by placing the vectors as columns.

If  \(\det(A)\neq 0\), then \(\mathbf{v}_1,...,\mathbf{v}_n\) are linearly independent.
If  \(\det(A)=0\), then \(\mathbf{v}_1,...,\mathbf{v}_n\) are linearly dependent.
Conclusion

So, here's the first miniature guide for Specialist Maths! I totally underestimated the workload I had in semester 1, and so I apologise for not contributing anything to the thread. It's the start of semester 2 now, and since I did a summer subject earlier this year, I get to underload, which means I \(^1\)should have a bit more time on my hands.

\(^1\) We'll see of course...
To request a topic, reply to this thread or send me a message!

[Tau]

Just to add one small note to the excellent guide by AlphaZero: any set of vectors containing the zero vector is linearly dependent.