1. EECS 1541 -- Introduction to Computing for the Physical Sciences
Week 2: Vectors and Matrices (Part I)
READING: – 2.1.2

2. EECS 1541 -- Introduction to Computing for the Physical Sciences
Arrays
Vectors and matrices are used to store set of values.
Like an array, vectors and matrices can be visualized as a table of values.
The dimensions of an array are r by c, where r is the number of rows and c is the number of columns.

3. EECS 1541 -- Introduction to Computing for the Physical Sciences
 returns an n-by-n matrix of pseudorandom normal values
EECS Introduction to Computing for the Physical Sciences
Scalars
A scalar in MATLAB is an array of size 1 x 1
Example:
a variable that stores a single value is an example of a scalar
>> z = 6
z = 6

4. EECS 1541 -- Introduction to Computing for the Physical Sciences
 returns an n-by-n matrix of pseudorandom normal values
EECS Introduction to Computing for the Physical Sciences
Vectors
A vector in MATLAB is equivalent to a 1-dimensional array where one of the dimensions is 1
There are: row vectors and column vectors
row vector
column vector

5. EECS 1541 -- Introduction to Computing for the Physical Sciences
 returns an n-by-n matrix of pseudorandom normal values
EECS Introduction to Computing for the Physical Sciences
Row Vectors
A row vector in MATLAB is an 1 x n array
Example of an 1 x 4 array that consists of 4 elements:
>> v = [ ]
v =
>> v = [1, 2, 3, 4]
v =

6. EECS 1541 -- Introduction to Computing for the Physical Sciences
 returns an n-by-n matrix of pseudorandom normal values
EECS Introduction to Computing for the Physical Sciences
Row Vectors
MATLAB workspace after a row vector has been assigned:

7. Row Vectors: Colon Operator
 returns an n-by-n matrix of pseudorandom normal values
EECS Introduction to Computing for the Physical Sciences
Row Vectors: Colon Operator
The colon operator can be used to iterate values in a vector
Format:
Vector_variable_name = initial value : step: final value
step size
Example 1:
>> v = 1:2:7
v =

8. Row Vectors: Colon Operator
 returns an n-by-n matrix of pseudorandom normal values
EECS Introduction to Computing for the Physical Sciences
Row Vectors: Colon Operator
Example 2:
If “step” is omitted, the default step size is 1
>> v = 1:6
v =

9. Row Vectors: Colon Operator
 returns an n-by-n matrix of pseudorandom normal values
EECS Introduction to Computing for the Physical Sciences
Row Vectors: Colon Operator
Example 3:
>> v = 1:3:12
v =
Adding 3 to 10 would go beyond 12, so the vector stops at 10

10. Row Vectors: Colon Operator
 returns an n-by-n matrix of pseudorandom normal values
EECS Introduction to Computing for the Physical Sciences
Row Vectors: Colon Operator
Example 4:
>> start = 20;
>> step = -3;
>> final = 6;
v = start:step:final
v =
Observe that the final value is not guaranteed to be at the end of the vector

11. EECS 1541 -- Introduction to Computing for the Physical Sciences
 returns an n-by-n matrix of pseudorandom normal values
EECS Introduction to Computing for the Physical Sciences
Row Vectors: Linspace
linspace(x,y,n) function creates a vector with n values in the inclusive range from x to y. It creates a vector with n values that are linearly spaced.
If n is omitted, the default is 100 points
If n = 1, linspace(x,y,1) returns y
Example :
>> v = linspace(1,5,3)
v =

12. Concatenating the vectors
 returns an n-by-n matrix of pseudorandom normal values
EECS Introduction to Computing for the Physical Sciences
Concatenating the vectors
Vectors variables can be created using existing variables
Example :
>> v1 = 2:5;
>> v2 = 1:3;
>> v_total = [v1 v2]
v_total =
From v1
From v2

13. EECS 1541 -- Introduction to Computing for the Physical Sciences
 returns an n-by-n matrix of pseudorandom normal values
EECS Introduction to Computing for the Physical Sciences
Column Vectors
A column vector in MATLAB is a n x 1 array
A column vector can be created directly by entering the values of the vector inside a pair of square brackets with the values separated by semi-colons
Example :
>> v = [2; 4; 6; 8]
v = 2 4 6 8

14. EECS 1541 -- Introduction to Computing for the Physical Sciences
 returns an n-by-n matrix of pseudorandom normal values
EECS Introduction to Computing for the Physical Sciences
Column Vectors
Any row vector created using any method can be transposed to result in a column vector
Example :
>> r_v = 2:5;
>> c_v = r_v’
c_v = 2 3 4 5

15. EECS 1541 -- Introduction to Computing for the Physical Sciences
 returns an n-by-n matrix of pseudorandom normal values
EECS Introduction to Computing for the Physical Sciences
Column Vectors
A column vector can also be created using the colon notation like the following example:
Example :
>> v = 1:5;
>> c = v(:)
c = 1 2 3 4 5

16. Number of elements in a vector
 returns an n-by-n matrix of pseudorandom normal values
EECS Introduction to Computing for the Physical Sciences
Number of elements in a vector
The function length will return the number of elements in a vector (i.e. both row vectors and column vectors)
Example :
>> v = 2:3:9
v =
>> length(v)
ans = 3

17. Number of elements in a vector
 returns an n-by-n matrix of pseudorandom normal values
EECS Introduction to Computing for the Physical Sciences
Number of elements in a vector
Example :
>> v = 3:7;
>> w = v’’;
>> length(w)
ans = 5

18. Indexing elements of a vector
 returns an n-by-n matrix of pseudorandom normal values
EECS Introduction to Computing for the Physical Sciences
Indexing elements of a vector
The elements in a vector are numbered sequentially; like an array, each element number is called the index
In MATLAB, the indices start at 1
index
element
To access a particular element in a vector, enter the name of the vector variable and the index in parentheses
vector_name()

19. Indexing elements of a vector
 returns an n-by-n matrix of pseudorandom normal values
EECS Introduction to Computing for the Physical Sciences
Indexing elements of a vector
Example :
>> v = -10:2:0
v =
>> v(2)
ans =
index is 2, so get the value of the second element in v -8

20. Indexing elements of a vector
 returns an n-by-n matrix of pseudorandom normal values
EECS Introduction to Computing for the Physical Sciences
Indexing elements of a vector
Example :
>> v = -10:2:0
v =
>> v(5)
ans = -2
index is 5, so get the value of the fifth element in v

21. Indexing elements of a vector
 returns an n-by-n matrix of pseudorandom normal values
EECS Introduction to Computing for the Physical Sciences
Indexing elements of a vector
Example :
>> v = -10:2:0
v =
>> v(end)
ans =
“end” refers to the last element in the vector

22. Indexing elements of a vector
 returns an n-by-n matrix of pseudorandom normal values
EECS Introduction to Computing for the Physical Sciences
Indexing elements of a vector
Example :
>> v = -10:2:0
v =
>> v(end-1)
ans = -2

23. Indexing elements of a vector
 returns an n-by-n matrix of pseudorandom normal values
EECS Introduction to Computing for the Physical Sciences
Indexing elements of a vector
Example :
>> v = -10:2:0
v =
>> v(2:4)
ans =
You can return a range of elements by inputting the indices

24. Indexing elements of a vector
 returns an n-by-n matrix of pseudorandom normal values
EECS Introduction to Computing for the Physical Sciences
Indexing elements of a vector
Example :
>> v = -10:2:0
v =
>> v([1 3])
ans =
The index can be a vector of indices
In this example, we want to obtain the first and the third element of the vector

25. Indexing elements of a vector
 returns an n-by-n matrix of pseudorandom normal values
EECS Introduction to Computing for the Physical Sciences
Indexing elements of a vector
Example :
>> a = linspace(2,6,5);
>> b = a’;
>> b(length(b)-3)
ans = 3
Based on this example, can you think of a general formula on how MATLAB compute all the values in linspace (when n is not equal to 1)?

26. EECS 1541 -- Introduction to Computing for the Physical Sciences
 returns an n-by-n matrix of pseudorandom normal values
EECS Introduction to Computing for the Physical Sciences
Magnitude of a vector
Magnitude of a vector (v) in n-dimensional can be computed by:
Magnitude of a vector can be computed using the norm function
For example, the magnitude of the vector (1,1,1) can be computer in MATLAB:
>> v = [1 1 1];
>> v_mag = norm(v)
v_mag =
1.7321

27. EECS 1541 -- Introduction to Computing for the Physical Sciences
 returns an n-by-n matrix of pseudorandom normal values
EECS Introduction to Computing for the Physical Sciences
Magnitude of a vector
You can select the elements in a vector and find out the corresponding magnitude using the norm function
>> v = [1 1 1];
Example,
>> v = [1 1 1];
>> v_mag = norm(v(1:2))
v_mag =
1.4142