Vectors in Julia
Reese Pathak
Keegan Go
David Zeng
EE103
Stanford University
September 28, 2016
Stephen Boyd
Vectors in Julia
main topics:
I
how to create and manipulate vectors in Julia
I
how Julia notation differs from math notation
2
Outline
Vectors
Vector operations
Norm and distance
Vectors
3
Vectors
I
vectors are represented by arrays in Julia
I
to create the 3-vector


8
x = (8, −4, 3.5) =  −4 
3.5
use
x = [8, -4, 3.5]
I
(x = [8;-4;3.5] also works)
watch out for similar looking expressions
I
length of a vector: length(x)
– (8,-4,3.5) and [8 -4 3.5] are not equivalent in Julia
Vectors
4
Ranges
I
to get a range from i to j (for i ≤ j), use a colon (:)
– the range from 1 to 10 is 1:10
– collect(1:10) returns the array
I
I
the default increment between values is 1. (1:3 is 1, 2, 3)
to specify an increment size add an additional argument:
– the range from 1 to 10 with a step size of 0.1 is 1:0.1:10
Vectors
5
Indexing and slicing
I
indexes run from 1 to n: x2 is x[2]
I
can also set an element, e.g., x[3] = 10.5
use a range to select more than one element
I
– x[2:3] selects the second and third elements
I
x[end] selects the last element
I
to select every other element use x[1:2:end]
Vectors
6
Block vectors
I
to form a stacked vector like
a = (b, c) =
b
c
(with b and c vectors)
a = [b; c]
(a = [b, c] does NOT work)
I
can mix vectors and scalars:
a = [b; 2; c; -6]
Vectors
7
Basic functions for arrays
I
sum of (the entries of) a vector: sum(x)
I
mean of the entries (avg(x)): mean(x)
I
0n is zeros(n)
I
1n is ones(n)
Vectors
8
Creating unit vectors
I
form e3 with length 10
I
create a zero vector of size 10 then set the third element to 1
e_3 = zeros(10); e_3[3] = 1;
Vectors
9
List of vectors
I
to form a list with vectors a, b, and c:
vector_list = [a,b,c]
I
the second vector in this list is vector_list[2]
I
to access an element in a vector: vertor_list[2][3]
Vectors
10
Outline
Vectors
Vector operations
Norm and distance
Vector operations
11
Vector addition and subtraction
I
vector addition uses +, for example

 

1
4
 2 + 5 
3
6
is written
[1, 2, 3] + [4, 5, 6]
I
subtraction uses -
I
the arrays must have the same length (unless one is scalar)
Vector operations
12
Scalar-vector addition
I
in Julia, a scalar and a vector can be added
I
the scalar is added to each entry of the vector
[2, 4, 8] + 3
gives (in mathematical notation)




2
5
 4  + 31 =  7 
8
11
Vector operations
13
Scalar-vector multiplication
I
scalar-vector multiplication uses *
I
for example,


1
(−2)  9 
6
is written
-2 * [1, 9, 6]
I
the other order gives the same result:
[1, 9, 6] * -2
Vector operations
14
Inner product
I
inner product aT b is written as
dot(a,b)
I
a and b must have the same length
Vector operations
15
Outline
Vectors
Vector operations
Norm and distance
Norm and distance
16
Norm and distance
I
the norm kxk =
p
x21 + x22 + · · · + x2n is written
norm(x)
I
dist(x, y) = kx − yk is written
norm(x-y)
Norm and distance
17
RMS value
I
rms(x) is defined as
r
rms(x) =
I
1 2
kxk
(x + · · · + x2n ) = √ .
n 1
n
can be expressed as
rms_x = norm(x)/sqrt(length(x))
Norm and distance
18
Standard deviation
I
standard deviation is defined as
std(x) =
I
kx − avg(x)1k
√
n
which can be expressed as
std_of_x = norm(x - mean(x))/sqrt(length(x))
I
warning: the Julia function std does not use this definition
Norm and distance
19
Angle
I
the angle between two vectors a and b is
T a b
∠(a, b) = arccos
kak kbk
I
can be expressed as
angle_a_b = acos(dot(a,b)/(norm(a)*norm(b)))
Norm and distance
20