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====== Math 445 lecture 2: Vectors ======

topics

<code>
[]       % creating vector vectors with square brackets
()       % accessing/assigning vector elements with parentheses
'        % the transpose operator, apostrophe
:        % the colon operator, making range vectors
+ -      % vector addition, subtraction
.* ./ .^ % elementwise vector operations
</code>
===== Vectors in math =====

Vectors are hugely important in math, science, engineering.
In math, a vector is a list of numbers. E.g. this is a 3-d vector 

<latex>
  x = \left(\begin{array}{c} 4.3 \\ 5.9 \\ 0.1 \end{array} \right)
</latex>


In math, the components or elements of a vector are written with subscripts.
E.g. for the above vector

<latex>
 x_1 = 4.3, \quad x_2 = 5.9, \quad x_3 = 0.1
</latex>

===== Creating vectors with explicit lists in square brackets =====
In Matlab, there's a distinction between row vector and column vectors.
You can construct **row vectors** by listing elements between **square brackets**, 
with either **spaces** or **commas** between.

<code>
>> x = [4.3 5.9 0.1]  % assign a row vector into variable x

x =

    4.3000    5.9000    0.1000 
</code>


To create a column vectors, put semicolons between the elements.

<code>
 >> x = [4.3; 5.9; 0.1] % assign a col vector into variable v

x =

    4.3000
    5.9000
    0.1000
</code>

===== Accessing elements with parentheses =====

In Matlab, you access the elements of a vector using **parentheses**. E.g. given the above vector $x$, you **get** the 2nd component of $x$ this way

<code>
>> x(2)

ans =

    5.9000
</code>

Note what happened there. We asked Matlab for the 2nd element of x. It **returned** the value 5.9 and assigned it to the default return variable ''ans''. Alternatively, the following will **assign** or **set** the value of the second component of $x$ to another number.

<code>
>> x(2) = 2.7

x =
    4.3000
    2.7000
    0.1000
</code>

===== The transpose operator ' (apostrophe) =====

Note that last method. In Matlab, the apostrophe '' ' '' stands for the **transpose**.
The transpose operator turns a row vector into a column vector, and vice versa.


<code>
>> x = [4.3 5.9 0.1]

x =

    4.3000    5.9000    0.1000

>> x'

ans =

    4.3000
    5.9000
    0.1000

>> y = x'

y =

    4.3000
    5.9000
    0.1000

>> z = y'

z =

    4.3000    5.9000    0.1000

</code>

===== Constructing vectors with the colon operator : =====

The Matlab **colon** operator : is used in a number of ways. Here we'll see how it can be used to create vectors. The most straightforward of these is the syntax ''m:n'', which creates a row vector of the integers from ''m'' to ''n''. E.g.

<code>
>> 1:10

ans =

     1     2     3     4     5     6     7     8     9    10

>> x = 4:7

x =

     4     5     6     7

</code>

You can also specify a **step** or **increment** between the elements using the syntax ''m:step:n''. For example, ''5:2:17'' will produce a vector of odd integers between 5 and 17.

<code>
>> x = 5:2:17

x =

     5     7     9    11    13    15    17
</code>

Lastly, the colon operator works with nonintegers, too. E.g.

<code>
>> x = 0:0.1:0.7

x =

         0    0.1000    0.2000    0.3000    0.4000    0.5000    0.6000    0.7000 
</code>

===== Constructing vectors with linspace : =====

In Matlab, vectors are often used to represent *discretized functions*, that is, the values of a function at a discrete set of gridpoints. The starting place for this is often the **linspace** command:

<code matlab>
x = linspace(0, 2*pi, 100); % construct a vector x of 100 gridpoints evenly spaced between 0 and 2pi

f = sin(x);                 % construct a vector f of sin evaluated at the gridpoints x

whos x f
  Name      Size             Bytes  Class     Attributes

  f         1x100              800  double              
  x         1x100              800  double              
</code>

Note that 'f' **is not actually a function**. It's just a vector of the values of $sin(x)$ at the 100 points in the vector 'x'. You can see that 'f' is a vector by the output of 'whos x f', which shows that 
the variables 'x' and 'f' are both size '1 x 100', i.e. row vectors of length 100. Note also 
the use of the semicolon to supress output (who wants to read all 100 elements of f and x).

===== Supplemental material =====

[[https://www.youtube.com/watch?v=zXP_pr7np-o | ' (apostrophe), Frank Zappa (the song)]]


[[https://www.youtube.com/watch?v=kQDp2425WQ0&list=PL8WvZFiJpAr29HFJkRcE4NvWCeYy3ytte | ' (apostrophe), Frank Zappa (full album) ]]
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