Posts Tagged ‘math1231’

MATH1081 and MATH1231 Cheat Sheets

March 19, 2009 Leave a comment

Just cross posting my cheat sheets from MATH1081 and MATH1231. PDF and LaTeX source ( won’t allow .tex uploads) are provided. These are from 08s2, at UNSW. I’ve released them into the public domain, if that is not legally possibly in Australia then lobby your parliament representative.




MATH1081 LaTeX Source

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            pdfauthor={Andrew Harvey},
            pdftitle={MATH1081 Cheat sheet 2008},


\subsection*{Counting Methods}
Let $\#(n)$ denote the number of ways of doing $n$ things. Then,
$$\#(A\; \mbox{and}\; B) = \#(A) \times \#(B)$$
$$\#(A\; \mbox{or}\; B) = \#(A) + \#(B)$$

($n$ items, $r$ choices)\\
Ordered selection with repetition, $n^r$.\\

Ordered selection without repetition, $P(n,r) = \frac{n!}{(n-r)!}$.\\

Unordered selection without repetition, $C(n,r) = \frac{P(n,r)}{r!}$.\\

$|A \cup B| = |A| + |B| - |A \cap B|$\\

Ordered selection with repeition; WOOLLOOMOOLOO problem.\\

Unordered selection with repetition; dots and lines,

Pigeonhole principle. If you have n holes and more than n objects, then there must be at least 1 hole with more than 1 object.

\subsection*{Formal Languages}
$\lambda$ represents the \textit{empty word}.
$w$ is just a variable (it is not part of the language)

\subsection*{First Order Homogeneous Case}
The recurrence,\\
\begin{center}$a_n = ra_{n-1}$ with $a_0=A$\\\end{center}
has solution\\

\subsection*{Second Order Recurrences}
$$a_n + pa_{n-1} + qa_{n-2} = 0$$
has characteristic,
$$r^2 + pr + q = 0$$
If $\alpha$ and $\beta$ are the solutions to the characteristic equation, and if they are real and $\alpha \ne \beta$ then,
$$a_n = A\alpha^n + B\beta^n.$$
If $\alpha = \beta$ then,
$$a_n = A\alpha^n + Bn\beta^n.$$

\subsection*{Guesses for a particular solution}
\begin{tabular}{c|c}%{m{width} | m{width}}
 LHS & Guess \\ \hline
$3$ & c \\
$3n$ & $cn + d$\\
$3\times 2^n$ & $c2^n$\\
$3n2^2$ & $(cn+d)2^n$\\
$(-3)^n$ & $c(-3)^n$\\


MATH1231-ALG LaTeX Source

%This work is hereby released into the Public Domain. To view a copy of the public domain dedication, visit or send a letter to Creative Commons, 171 Second Street, Suite 300, San Francisco, California, 94105, USA.


            pdfauthor={Andrew Harvey},
            pdftitle={MATH1231 Algebra Cheat sheet 2008},


\section*{Vector Spaces}
\subsection*{Vector Spaces}
Vector Space - 10 rules to satisfy, including \begin{small}\textit{(Where $V$ is a vector space.)}\end{small}
\item Closure under addition. If $\mathbf{u}, \mathbf{v} \in V$ then $\mathbf{u} + \mathbf{v} \in V$
\item Existence of zero. $\mathbf{0} \in V$
\item Closure under scalar multiplication. If $\mathbf{u} \in V$ then $\lambda \mathbf{u} \in V$, where $\lambda \in \mathbb{R}$

Subspace Theorem:\\
  A subset $S$ of a vectorspace $V$ is a subspace if:
\item $S$ contains the zero vector.
\item If $\mathbf{u}, \mathbf{v} \in S$ then $\mathbf{u} + \mathbf{v} \in S$ and $\lambda \mathbf{u} \in S$ for all scalars $\lambda$.

\subsection*{Column Space}
The column space of a matrix $A$ is defined as the span of the columns of $A$, written $\mbox{col}(A)$.

\subsection*{Linear Independence}
Let $S = \{\mathbf{v_1}, \mathbf{v_2}, \dots, \mathbf{v_n}\}$ be a set of vectors.
\item If we can find scalars $\alpha_1 + \alpha_2 + \dots + \alpha_n$ not all zero such that
\begin{center}$\alpha_1\mathbf{v_1} + \alpha_2\mathbf{v_2} + \dots + \alpha_n\mathbf{v_n} = 0$\end{center}
then we call $S$ a linearly dependent set and that the vectors in $S$ are linearly dependent.

\item If the only solution of
\begin{center}$\alpha_1\mathbf{v_1} + \alpha_2\mathbf{v_2} + \dots + \alpha_n\mathbf{v_n} = 0$\end{center}
is $\alpha_1 = \alpha_2 = \dots = \alpha_n = 0$ then $S$ is called a linearly independent set and that the vectors in $S$ are linearly independent.

A set $B$ of vectors in a vectorspace $V$ is called a basis if
\item $B$ is linearly independent, and
\item $V = \mbox{span}(B)$.

An orthonormal basis is formed where all the basis vectors are unit length and are mutually orthogonal (the dot product of any two is zero).

The dimension of a vectorspace $V$, written dim($V$) is the number of basis vectors.

\section*{Linear Transformations}
\subsection*{Linear Map}
A function $T$ which maps from a vectorspace $V$ to a vectorspace $W$ is said to be linear if, for all vectors $\mathbf{u}, \mathbf{v} \in V$ and for any scalar $\lambda$,
\item $T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v})$,
\item $T(\lambda\mathbf{u}) = \lambda T(\mathbf{u})$.

The columns of the transformation matrix are simply the images of the standard basis vectors.

\subsection*{The Kernel}
$\mbox{im}(T) = \mbox{col}(A)$\\
The kernel of a linear map $T : V \to W$, written ker($T$), consists of the set of vectors $\mathbf{v} \in V$ such that $T(\mathbf{v}) = \mathbf{0}$.\\
If A is the matrix representation of a linear map $T$, then the kernel of $A$ is the solution set of $A\mathbf{x} = \mathbf{0}$.

The dimension of the image of a linear map $T$ is called the rank of $T$, written rank($T$). (Maximum number of linearly independent columns of A)\\

The dimension of the kernel of a linear map $T$ is called the nullity of $T$, written nullity($T$). (Number of parameters in the solution set of $A\mathbf{x}=\mathbf{0}$)\\

If $T$ is a linear map from $V$ to $W$ then
$$\mbox{rank}(T) + \mbox{nullity}(T) = \mbox{dim}(V)$$

\section*{Eigenvectors and Eigenvalues}
If $A$ is a square matrix, $\mathbf{v} \ne 0$ and $\lambda$ is a scalar such that,
$$A\mathbf{v} = \lambda\mathbf{v}$$
then $\mathbf{v}$ is an eigenvector of $A$ with eigenvalue $\lambda$.\\
A\mathbf{v} = \lambda\mathbf{v}
\implies (A-\lambda I)\mathbf{v} = \mathbf{0}

A-\lambda I \mbox{has no inverse (otherwise } \mathbf{v} = \mathbf{0} \mbox{)}
\mbox{so set det}(A-\lambda I) = 0
 \mbox{this finds the eigenvectors}

\mbox{Also since}
(A-\lambda I)\mathbf{v} = \mathbf{0}
\mathbf{v} \in \mbox{ker}(A-\lambda I)
\mbox{this gives eigenvectors.}
Eigenvalues: Set $\mbox{det}(A-\lambda I) = 0$ and solve for $\lambda.$\\
Eigenvectors: For each value of $\lambda$ find the kernel of $(A-\lambda I)$.

If $A$ has $n$ (independent) eigenvectors then put,\\
$P = (\mathbf{v_1}|\mathbf{v_2}|...|\mathbf{v_n})$ (eigenvectors $\mathbf{v}$)\\
$D =
  \lambda_1 &        & 0         \\
            & \ddots &           \\
  0         &        & \lambda_n
$ (eigenvalues $\lambda$)\\
so then\\
$A^k = PD^kP^{-1}$, for each non-negative integer $k$.\\
Remember that when $A =
\bigl( \begin{smallmatrix}
  a&b\\ c&d
\end{smallmatrix} \bigr)$, $A^{-1} = \frac{1}{det(A)}
d & -b\\
-c & a\end{pmatrix}$

\subsection*{Systems of Differential Equations}
The system
	\frac{dx}{dt} = 4x + y\\
	\frac{dy}{dt} = x + 4y
 can be written $\mathbf{x}'(t) = \begin{pmatrix} 4 & 1\\1 & 4\end{pmatrix}\mathbf{x}(t)$ where $\mathbf{x}(t) = \binom{x(t)}{y(t)}.$

We can guess the solution to be $\mathbf{x}(t) = \alpha \mathbf{v} e^{\lambda t}$ (and add for all the eigenvalues). Where $\mathbf{v} $ and $ \lambda$ are the eigenvectors and eigenvalues respectively.

\section*{Probability and Statistics}

Two events $A$ and $B$ are mutually exclusive if $A \cap B = \emptyset$.
$$P(A^c) = 1 - P(A)$$
$$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$

Two events $A$ and $B$ are physically independent of each other if the probability that one of them occurs is not influenced by the occurrence or non occurrence of the other. These two events are statistically independent if, $$P(A \cap B) = P(A).P(B).$$

\subsection*{Conditional Probability}
Probability of $A$ given $B$ is given by,
$$P(A|B) = \frac{P(A \cap B)}{P(B)}$$

\subsection*{Bayes Rule}

\subsection*{Discrete Random Variables}
$$p_k = P(X=k) \qquad \mbox{(} \{p_k\}\mbox{ is the probability distribution)}$$
where, $X$ is a discrete random variable, and $P(X=k)$ is the probability that $X=k$.\\

For $\{p_k\}$ to be a probability distribution,
\item $p_k \ge 0$ for $k = 0, 1, 2, \dots$
\item $p_0 + p_1 + \dots = 1$

\subsection*{Mean and Variance}
$E(X)$ denotes the mean or expected value of X.\\
%$$E(X) = \sum_{i=1}^{k} p_i x_i \qquad  $$
$$E(X) = \sum_{\mbox{all }k} kp_k$$

$$\mbox{Var}(X) = E(X^2) - E(X)^2 \qquad \mbox{where } E(X^2) = \sum_{\mbox{all } k} k^2 p_k$$

\subsection*{Binomial Distribution}
If we perform a binomial experiment (i.e. 2 outcomes) $n$ times, and each time there is a probability $p$ of success then,
$$P(X=k) = \binom{n}{k} p^k (1-p)^{n-k},\qquad \mbox{for } 0\le k \le n \mbox{ and 0 otherwise}.$$

\subsection*{Geometric Distribution}
$$P(X=k) = p(1-p)^{k-1}, \;\; k = 1, 2, \dots$$

MATH1231-CACL LaTeX Source

%This work is hereby released into the Public Domain. To view a copy of the public domain dedication, visit or send a letter to Creative Commons, 171 Second Street, Suite 300, San Francisco, California, 94105, USA.


pdfauthor={Andrew Harvey},
pdftitle={MATH1231 Calculus Cheat sheet 2008},

\section*{Functions of Several Variables}
\item Level Curves
\item Sections

\subsection*{Partial Derivatives}
$$\frac{\partial^2 f}{\partial y \partial x} = \frac{\partial^2 f}{\partial x \partial y}$$
on every open set on which $f$ and the partials,\\
$$\frac{\partial f}{x}, \frac{\partial f}{y}, \frac{\partial^2 f}{\partial y \partial x}, \frac{\partial^2 f}{\partial x \partial y}$$
are continuous.

\subsection*{Normal Vector}
$$\mathbf{n} = \pm
\frac{\partial f(x_0, y_0)}{\partial x}\\
\frac{\partial f(x_0, y_0)}{\partial y}\\

\subsection*{Error Estimation}
$$f(x_0 + \Delta x, y_0 + \Delta y) \approx f(x_0, y_0) + \left[ \frac{\partial f}{\partial x} (x_0, y_0)\right] \Delta x + \left[ \frac{\partial f}{\partial y} (x_0, y_0)\right] \Delta y$$

\subsection*{Chain Rules}
Example, $z = f(x,y)\ \mbox{with}\ x = x(t), y = y(t)$
& f \ar[dl] \ar[dr] &   \\
x \ar[d] &                   & y \ar[d]\\
t        &                   & t}

$$\frac{df}{dt} = \frac{\partial f}{\partial x} \cdot \frac{dx}{dt} + \frac{\partial f}{\partial y} \cdot \frac{dy}{dt}$$

\subsection*{Integration by Parts}
Integrate the chain rule,
$$u(x)v(x) = \int u'(x)v(x)\; dx + \int v'(x)u(x)\; dx$$

\subsection*{Integration of Trig Functions}
For $\int \sin^2 x\; dx$ and $\int \cos^2 x\; dx$ remember that,\\
$$\cos 2x = \cos^2 x - \sin^2 x$$
%$$\sin 2x = 2\sin x \cos x$$

Integrals of the form $\int \cos^m x \sin^n x\; dx$, when $m$ or $n$ are odd, you can factorise using $\cos^2 x + \sin^x = 1$  and then using,
$$\int \sin^k x \cos x\; dx = \frac{1}{k+1} \sin^{k+1} x + C$$
$$\int \cos^k x \sin x\; dx = \frac{-1}{k+1} \cos^{k+1} x + C$$

\subsection*{Reduction Formulae}

\subsection*{Partial Fractions}
Example, assume
$$\frac{2x-1}{(x+3)(x+2)^2} = \frac{A}{x+3} + \frac{Bx + C}{(x+2)^2}$$
\item $\frac{2x-1}{x+2} = A + \frac{B(x+3)}{x+2}$ subs $x=-3 \to A=7$
\item $\frac{2x-1}{x+3} = \frac{A(x+2)}{x+3} + B$ subs $x=-2 \to B=-5$
Now multiply both sides by $(x+2)(x+3)$ and equate coefficients.

\subsection*{Separable ODE}
Separate then integrate.

\subsection*{Linear ODE}
The ODE:
$$ \frac{dy}{dx} + f(x)y=g(x) $$
has solution,
$$ y(x) = \frac{1}{u(x)}\left [ \int u(x)g(x)\ dx + C \right] $$
$$ u(x) := e^{\int f(x)\ dx} $$

\subsection*{Exact ODE}
The ODE:
$$ \frac{dy}{dx} = -\frac{M(x,y)}{N(x,y)} $$
or as,
$$ M(x,y)dx + N(x,y)dy = 0 $$
is exact when,
$$ \frac{\partial M}{\partial y} = \frac{\partial N}{\partial x} $$

Assume solution is of the form,
$$ F(x,y) = c $$
$$ M = \frac{\partial F}{\partial x} \qquad N = \frac{\partial F}{\partial y}$$

Integrate to find two equations equal to $F(x,y)$, then compare and find solution from assumed form.

\subsection*{Second Order ODE's}
The ODE:
$$ay'' + ay' + by = f(x)$$
For the homogeneous case ($f(x) = 0$)\\
the characteristic equation will be $a\lambda^2 + b\lambda + c = 0$

If the characteristic equation has,\\
Two Distinct Real roots, (replace the $\lambda$'s with the solutions to the characteristic eqn.)
$$ y = Ae^{\lambda x} + Be^{\lambda x}$$

Repeated Real roots,
$$ y = Ae^{\lambda x} + Bxe^{\lambda x}$$

Complex roots,
$$ y = e^{\alpha x}(A\cos \beta x + B\sin \beta x)$$

$$\lambda = \alpha \pm \beta i$$

For the For the homogeneous case,
$$y = y_h + y_p$$
$$y = \mbox{solution to homogeneous case} + \mbox{particular solution}$$

Guess something that is in the same form as the RHS.\\
If $f(x) = P(x)\cos ax \mbox{(or sin)}$ a guess for $y_p$ is $Q_1(x)\cos ax + Q_2(x)\sin ax$

\section*{Taylor Series}
\subsection*{Taylor Polynomials}
For a differentiable function $f$ the Taylor polynomial of order $n$ at $x=a$ is,
$$P_n(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \dots + \frac{f^{\left( n\right) }(a)}{n!}(x-a)^n$$

\subsection*{Taylor's Theorem}
$$f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \dots + \frac{f^{\left( n\right) }(a)}{n!}(x-a)^n + R_n(x)$$

$$R_n(x) = \frac{f^{\left( n+1\right) }(c)}{(n+1)!}(x-a)^{n+1}$$

$$\lim_{x \to \infty} f(x) = L \implies \lim_{n \to \infty}a_n = L$$
essentially says that when evaluating limits functions and sequences are identical.

A sequence diverges when $\displaystyle \lim_{n \to \infty}a_n = \pm \infty$ or $\displaystyle \lim_{n \to \infty}a_n$ does not exist.

\subsection*{Infinite Series}
\subsubsection*{Telscoping Series}
Most of the terms cancel out.

\subsubsection*{$n$-th term test (shows divergence)}
$\displaystyle \sum_{n=1}^{\infty} a_n$ diverges if $\displaystyle \lim_{n \to \infty}{a_n}$ fails to exist or is non-zero.

\subsubsection*{Integral Test}
Draw a picture. Use when you can easily find the integral.

\subsubsection*{$p$- series}
The infinite series $\displaystyle \sum_{n=1}^{\infty} \frac{1}{n^p}$ converges if $p>1$ and diverges otherwise.

\subsubsection*{Comparison Test}
Compare to a p-series.

\subsubsection*{Limit form of Comparison Test}
Look at $\displaystyle \lim_{n \to \infty}{\frac{a_n}{b_n}}\;$ where $b_n$ is usually a p-series.\\
If $=c>0$, then $\sum a_n$ and $\sum b_n$ both converge or both diverge.\\
If $=0$ and $\sum b_n$ converges, then $\sum a_n$ converges.\\
If $=\infty$ and $\sum b_n$ diverges, then $\sum a_n$ diverges.

\subsubsection*{Ratio Test}
$$\lim_{n \to \infty} \frac{a_{n+1}}{a_n} = \rho$$

The series converges if $\rho < 1$,\\
the series diverges if $\rho > 1$ or $\rho$ is infinite,\\
and the test is inconclusive if $\rho = 1$.

\subsubsection*{Alternating Series Test}
The series,
$$\sum_{n=1}^{\infty} (-1)^{n+1} u_n = u_1 - u_2 + u_3 - u_4 + \dots$$
converges if,
\item The $u_n$'s are all $>0$,
\item $u_n \ge u_{n+1}$ for all $n \ge N$ for some integer $N$, and
\item $u_n \rightarrow 0$.

\subsubsection*{Absolute Convergence}
If $\displaystyle \sum_{n=1}^{\infty} |a_n|$ converges, then $\displaystyle \sum_{n=1}^{\infty} a_n$ converges.

\subsection*{Taylor Series}
Taylor Polynomials consist of adding a finite number of things together, whereas Taylor Series is an infinite sum.\\
The Maclaurin series is the Taylor series at $x=0$.
\subsection*{Power Series}

\section*{More Calculus}
\subsection*{Average Value of a Function}
$$\frac{\displaystyle \int_a^b f(x)\ dx}{b-a}$$

\subsection*{Arc Length}
Arc length over $\displaystyle[a,b] = \int_a^b \sqrt{1+f'(x)^2}\ dx$\\
$$s = \int_a^b \sqrt{x'(t)^2 + y'(t)^2}\ dt \qquad \mbox{(parametric)}$$

$$\frac{ds}{dt} = \sqrt{x'(t)^2 + y'(t)^2}$$

\subsection*{Surface Area of Revolution}
$$2\pi \int_a^b f(x) \sqrt{1+f'(x)^2}\ dx$$
$$2\pi \int_a^b y(t) \sqrt{x'(t)^2 + y'(t)^2}\ dt \qquad \mbox{(parametric)}$$
Categories: mathematics, unswcourse Tags: ,

Units, Vectors and Bases. Metrics?

October 2, 2008 Leave a comment

I began wondering about this mainly when the idea of a basis is used in linear algebra, although it seems to be strongly related to scalars, just as basis is to vector as one is to scalar.

This area of investigation for me arose when then they took coordinate vectors out of the UNSW MATH1231 (2007) syllabus. So I had to do a bit of reading on my own. To understand coordinate vectors its helpful to look at the vector space of polynomials. For example take the polynomial 3 – 2x + x2. Now if we put the coefficients into a vector, (3, -2, 1) then this vector is called the coordinate vector of 3 – 2x + x2 with respect to the ordered basis {1, x, x2}[1]. This sounds clear, but let me propose the following.

Say we have the coordinate vector (1,2,3) with respect the the ordered basis S = {(1,0,0), (0,1,0), (0,0,1)}, denoted [(1,2,3)]S. How is this different to just (1,2,3)? Aren’t all vectors, by convention defined with respect to the standard basis of that vector space? But that cannot possibly make sense because bases are defined in terms of vectors, so that would be a recursive definition which is not logically valid. There must be a more reasonably explanation.

Now looking at this in the vector space of say \mathbb{R}^2 we can take one basis of B = {(1,0), (0,1)} which is the standard basis, and another C = {(1,1), (-1,1)}.

Now if we define the point (1,2) with respect the the basis C, then with respect to the basis B this is the point, 1(1, 1) + 2(-1, 1) = (-1, 3). So essentially this allows us to still work with respect to the standard basis B, but we can work in the frame of C which allows us to then say treat the point (1,1) as just (1,0) which can simplify things.

Again looking at this graphically say in \mathbb{R}^2 then we can just draw essentially any two non-parallel lines which we call our axis or basis. Any mathematics that is done in any of these coordinate systems with respect to that coordinate system (i.e. the two lines drawn) would be the same. (1,1) + (2,0) would give (3,1) in both systems. Its only when we start defining one vector in one system with respect to the basis of the other system that we need to worry about the difference between the bases.

In the same sense I would say, mathematically it is meaningless to draw some Euclidean geometry on paper without drawing in your standard basis. Without them everything is meaningless as you define vectors as coordinate vectors with respect to some basis. Just as the polynomial 3 – 2x + x2 is really just the vector (3, -2, 1) with a standard basis of {1, x, x2}. I guess you would call them metrics of the vector space of \mathbb{R}^2, just like 1 is the metric of all subsets of the real numbers (is it?).

As a final note I think this is just the same as 2 is really means 2 × 1, and 3 means three lots of 1’s. Any links to abstract algebra? Probably, I’m not sure.

[1] MATH1231 Algebra [Online]. Angell D. 2005. Accessed July 2008. Ch 6, Pg 70.

Categories: Uncategorized Tags: ,

(x,y,z,w) in OpenGL/Direct3D (Homogeneous Coordinates)

September 29, 2008 7 comments

I always wondered why 3D points in OpenGL, Direct3D and in general computer graphics were always represented as (x,y,z,w) (i.e. why do we use four dimensions to represent a 3D point, what’s the w for?). This representation of coordinates with the extra dimension is know as homogeneous coordinates. Now after finally getting formally taught linear algebra I know the answer, and its rather simple, but I’ll start from the basics.

Points can be represented as vectors, eg. (1,1,1). Now a common thing we want to do in computer graphics is to move this point (translation). So we can do this by simply adding two vectors together,

\begin{pmatrix}x'\\y'\\z'\end{pmatrix} = \begin{pmatrix}x\\y\\z\end{pmatrix} + \begin{pmatrix}a\\b\\c\end{pmatrix} = \begin{pmatrix}x + a\\y + b\\z + c\end{pmatrix}.

If we wanted do some kind of linear transformation such as rotate about the origin, scale about the origin, etc, then we could just multiply a certain matrix with the point vector to obtain the image of the vector under that transformation. For example,

\begin{pmatrix}x'\\ y'\\ z' \end{pmatrix} = \begin{pmatrix}\cos \theta &-\sin \theta &0\\ \sin \theta &\cos \theta &0\\ 0&0&1\end{pmatrix} \begin{pmatrix}x\\ y\\ z\end{pmatrix}

will rotate the vector (x,y,z) by angle theta about the z axis.

However as you may have seen you cannot do a 3D translation on a 3D point by just multiplying a 3 by 3 matrix by the vector. To fix this problem and allow all affine transformations (linear transformation followed by a translation) to be done by matrix multiplication we introduce an extra dimension to the point (denoted w in this blog). Now we can perform the translation,

\mathbb{R}^2 : (x,y) \to (x+a, y+b)

by a matrix multiplication,

\begin{pmatrix}1 & 0 & a\\ 0 & 1 & b\\ 0 & 0 & 1\end{pmatrix} \begin{pmatrix}x\\ y\\ 1\end{pmatrix} = \begin{pmatrix}x + a\\ y + b\\ 1 \end{pmatrix}.

We need this extra dimension for the multiplication to make sense, and it allows us to represent all affine transformations as matrix multiplication.

Homogeneous coordinates. (2008, September 29). In Wikipedia, The Free Encyclopedia. Retrieved 04:33, September 29, 2008, from