Transcript ES100: Lecture 02 Variables and Arrays

Lecture 2
MATLAB fundamentals
Variables, Naming Rules,
Arrays (numbers, scalars, vectors, matrices),
Arithmetical Operations,
Defining and manipulating arrays
Variables and Arrays

What are variables?
– Variables are arrays of numbers.
– You name the variables (as the programmer)
and assign them numerical values.
– You execute the assignment command to
place the variable in the workspace memory
(memory is part of hardware used for storing
information).
– You are allowed to use the variable in
algebraic expressions, etc. once it is assigned.
Variable Naming Rules
Must begin with a LETTER
 May only contain letters, numbers and
underscores ( _ )
 No spaces or punctuation marks allowed!
 Only the first 63 characters are significant;
beyond that the names are truncated.
 Case sensitive (e.g. the variables a and A
are not the same)

Which variable names are valid?






12oclockRock
tertiarySector
blue cows
Eiffel65
red_bananas
This_Variable_Name_Is_Quite_Possibly_Too_Lo
ng_To_Be_Considered_Good_Practice_However
_It_Will_Work % (the green part is not part of
the recognized name)
Variable Naming Conventions

There are different ways to name variables. The
following illustrate some of the conventions used:
– lowerCamelCase
– UpperCamelCase
– underscore_convention

If a variable is a constant, some programmers use all
caps:
– CONSTANT

It does not matter which convention you choose to work
with; it is up to you.
Variables as Arrays
In MATLAB, a variable is stored as an array of
numbers. When appropriate, it is interpreted as a
scalar, vector or matrix.
scalar
vector
matrix
1×1
n × 1 or 1 × n
n×m
The size of an array is specified by the number of
rows and the number of columns in the array, with
the number of rows indicated first.
Scalars
Scalars are 1×1 arrays.
 They contain a single value, for example:

r = 6
height = 5.3
width = 9.07
Vectors

A vector is a list of numbers expressed as a 1
dimensional array.
 A vector can be n×1 or 1×n.

Columns are separated by commas (or spaces):
h = [1, 2, 3]

Rows are separated by semicolons:
v = [1; 2; 3]
Matrices
A matrix is a two
dimensional array of
numbers.
Rows

Columns

For example, this is a
4×3 matrix:
1
2
3
1
3.0
1.8
3.6
2
4.6
-2.0
21.3
3
0.0
-6.1
12.8
4
2.3
0.3
-6.1
m = [3.0, 1.8, 3.6; 4.6, -2.0, 21.3; 0.0,
-6.1, 12.8; 2.3, 0.3, -6.1]
Indexed-location of numbers in an
array
Each item in an array
is located in the
(row, column).
m(2,3)
ans =
21.3000
Rows

Columns
1
2
3
1
3.0
1.8
3.6
2
4.6
-2.0
21.3
3
0.0
-6.1
12.8
4
2.3
0.3
-6.1
Hands-on

Enter the following into MATLAB:
– Scalar:
a = 1
– Vectors:
b = [1, 0, 2]
c = [1 0 2]
– Matrix:
d = [5, 4, 3; 0, 2, 8]
Hands-on

Enter (input) the following matrix into MATLAB:
whiteRabbit
=
-7
21
6
2
32
0
-5
0
-18.5
Scalar Operations
Operation
Algebraic
Syntax
MATLAB
Syntax
Addition
a+b
a + b
Subtraction
a-b
a – b
Multiplication
a×b
a .* b
Division
a÷b
a ./ b
Exponentiation
ab
a .^ b
Array Operations

Arrays of numbers in MATLAB can be interpreted as
vectors and matrices if vector or matrix algebra is to be
applied. Recall that matrices are mathematical objects
that can be multiplied by the rules of matrices. To do
matrix multiplication, you need to use the standard *, /,
and ^ operators [without the preceding . (dot)]. They are
not for array multiplication, division and exponentiation.

To deal with arrays on an element-by-element level we
need to use the following array or dot-operators:
.*
,
./
and
.^
Array operations & dot-operators
.*
,
./
and
.^
Because scalars are equivalent to a 1×1
array, you can either use the standard or
the dot-operators when doing
multiplication, division and exponentiation
of scalars (i.e., of single numbers).
 It is okay for you to always use the dotoperators, unless you intend to perform
vector or matrix multiplication or division.

Array vs. Matrix Operations

Example:
x = [2, 1; 3, 4]
y = [5, 6; 7, 8]
z = x .* y
results in [10, 6; 21, 32]; this is array multiplication
z = x * y
results in [17, 20; 43, 50]; this is matrix multiplication
So, do NOT forget the dot when doing array
operations! (.* ./ .^)
Hierarchy of Operations
Just like in mathematics the operations are done in the
following order: Left to right doing what is in
Parentheses & Exponents first, followed by
Multiplication & Division, and then
Addition & Subtraction last.
An example:
c
c
c
c
c
=
=
=
=
=
2+3^2+1/(1+2)
2+3^2+1/(1+2)
2+3^2+1/(1+2)
2+3^2+1/(1+2)
2+3^2+1/(1+2)
1st
2nd
3rd
4th
5th
c
c
c
c
c
=
=
=
=
=
2+3^2+1/3
2+9+1/3
2+9+0.33333
11+0.33333
11.33333
Hands-on

Enter these two arrays into MATLAB:
a =
b =
10
2
6

5
9
8
5
0
8
1
0
1
0
0
1
2
0
0
Multiply, element-by-element, a × b.
– Since this is an array operation, the .*
multiplication operation is implied by the
request.
Defining & manipulating arrays

All variables in MATLAB are arrays!
– Single number array & scalar:
– Row array & row vector:
– Column array & column vector:
– Array of n rows x m columns & Matrix:
– Naming rules
– Indexed by (row, column)

1×1
1×n
nx1
n×m
Remark: vectors and matrices are special
mathematical objects, arrays are lists or
tables of numbers.
The equal sign assigns

Consider the command lines:
>>
>>
>>
>>

ax = 5;
bx = [1 2];
by = [3 4];
b = bx + by;
The equal sign (=) commands that the
number computed on the right of it is
input to the variable named on the left;
thus, it is an assignment operation.
Defining (or assigning) arrays

An array can be defined by typing in a list of numbers
enclosed in square brackets:
– Commas or spaces separate numbers.
A = [12, 18, -3]
or
A =
12
18
-3
A = [12 18 -3]
– Semicolons indicate a new row.
B = [2, 5, 2; 1 , 1, 2; 0, -2, 6]
B =
2
5
2
1
1
2
0
-2
6
Defining arrays continued

You can define an array in terms of another array:
C = [A; B]
C =
12
18
2
5
1
1
0
-2
-3
2
2
6
D = [C, C]
D =
12
18
2
5
1
1
0
-2
-3
2
2
6
12
2
1
0
18
5
1
-2
-3
2
2
6
Creating Zeros & Ones arrays

Create an array of zeros:
E = zeros(3,5)
E =
0
0
0
0
0
0

0
0
0
0
0
0
0
0
0
Create an array of ones:
F = ones(2,3)
F =
1
1
1
1
1
1
Note: Placing a single number inside either function will return an n × n array.
e.g. ones(4) will return a 4 × 4 array filled with ones.
Retrieving Values in an Array

Index – a number used to identify elements in an array
– Retrieving a value from an array:
G = [1, 2, 3; 4, 5, 6; 7, 8, 9]
G =
1
2
3
4
5
6
7
8
9
G(2,1)
ans = 4
G(3,2)
ans = 8
Changing Values in an Array

You can change a value in an element in an array with indexing:
A(2) = 5
A =
12

5
-3
You can extend an array by defining a new element:
A(6) = 8
A =
12
5
-3
0
0
8
– Notice how undefined values of the array are filled with zeros
Colon Operator

Colon notation can be used to define evenly spaced vectors in the
form:
first : last
H = 1:6
H =
1

2
3
4
5
6
The default spacing is 1, so to use a different increment, use the form:
first : increment : last
I = 1:2:11
I =
1
3
5
– The numbers now increment by 2
7
9
11
Extracting Data with the Colon
Operator

The colon represents an entire row or column when used
in as an array index in place of a particular number.
G =
1
4
7
G(:,1)
ans =
1
4
7
2
5
8
G(:,3)
ans =
3
6
9
3
6
9
G(2,:)
ans =
4
5
6
Extracting Data with the Colon
Operator Continued

The colon operator can also be used to extract a range
of rows or columns:
G =
1
4
7
G(2:3,:)
G =
4
7
2
5
8
5
8
3
6
9
6
9
G(1,2:3)
ans =
2
3
Manipulating Arrays

The transpose operator, an apostrophe,
changes all of an array’s rows to columns
and columns to rows.
J = [1 , 3, 7]
J =
1
3
J'
ans =
7
1
3
7
Manipulating Matrices Continued

The functions fliplr() and flipud() flip a
matrix left-to-right and top-to-bottom,
respectively.
– Experiment with these functions to see how
they work.
Hands-on exercise:
Create the following matrix using colon notation:

W =
1
10
6

–
5
18
2
1:5
The second row ranges from 10 to 18 in increments of 2
10:2:18
The third row ranges from 6 to 2 in increments of -1

–
4
16
3
The first row ranges from 1 to 5 in increments of 1

–
3
14
4
All three rows are evenly spaced

–
2
12
5
6:-1:2
All together:
 W = [1:5; 10:2:18; 6:-1:2]
Hands-on continued

Create the following matrix using colon notation:
X =
1.2
1.9
0


2.3
3.8
-3
3.4
5.7
-6
4.5
7.6
-9
5.6
9.5
-12
Transpose this matrix and assign it to variable Y.
>> Y = x’
Extract the 2nd row from Y and assign it to variable Z.
>> Z = Y(2,:)
Summary (1 of 2)
Naming a variable: Start with letter
followed by any combination of letters,
numbers and underscores (up to 63 of
these objects are recognized).
 Arrays are rows and columns of numbers.
 Array operations (element-by-element
operations with the dot-operators)
 Hierarchy of arithmetic operations.

Summary (2 of 2)
Command lines that assign variables
numerical values start with the variable
name followed by = and then the defining
expression
 An array of numbers is the structure of
variables in MATLAB. Within one variable
name, a set of numbers can be stored.
 Array, vector and matrix operations are
efficient MATLAB computational tools.
