Introduction to Vectors

Return to List of Lectures

Previous Lecture

Misc

Geneseo programming and data science club: first meeting of the semester Thursday, Feb. 2, 6:00 - 7:00 PM, Bailey 104. With pizza!

Experimental survey for giving me anonymous questions to answer in class or feedback about this course: “Questions and Suggestions” under “Quizzes” in Canvas.

Questions?

Lab: how to combine the probability of matching the powerball with the probability of matching the regular numbers?

What you need to calculate is the probability of matching the regular numbers and matching the powerball number.

Since those two events are independent (i.e., neither influences the likelihood of the other), this probability is just the product of the probabilities of the separate events: pr{regular} × pr{powerball}

Which in turn is the result calculated in step 2 times 1/p. In almost-Matlab, assuming your step 2 calculation is in a function named step2, this might look like step2(k,n) * 1/p

Vectors

Vectors from Arbitrary Data

Create row and column vectors of selected student’s heights (or other numbers). Find the average (with the mean function)

Relevant reading ideas:

• Write vectors enclosed in brackets, with commas between elements for a row vector, semicolons for column
• Can pick individual elements out with name( index )
• Transpose via single-quote

>> v = [ 1, 2, 3 ]
v =
     1     2     3
>> v'
ans =
     1
     2
     3
>> [ 10; 20; 30; 40 ]'
ans =
    10    20    30    40

>> v  = [14,16,18,24]
v =
    14    16    18    24
>> mean( v )
ans =
    18
>> v2 = v'
v2 =
    14
    16
    18
    24
>> mean( v2 )
ans =
    18

Vectors with Regular Patterns

Example 1: Find the sum of the first n natural numbers by creating a vector of numbers 1 through n and applying the sum function

Example 2: Find the sum of the first n odd numbers by creating a vector of n numbers evenly spaced between 1 and 2n-1 and applying the sum function

Relevant ideas from the reading:

• Patterns in a vector’s elements can be generated with [ start : increment : end ]
• Use [ 1 : n ] to generate [ 1, 2, 3, ..., n ]
• Can also use linspace( start, stop, number_of_points ) or logspace( start_power, stop_power, number_of_points )

>> % Sum of first n natural numbers
>> c = [ 1 : 7 ]
c =
     1     2     3     4     5     6     7
>> sum( c )
ans =
    28
>> sum( [1:500] )
ans =
      125250
>> % sum of first n odd numbers
>> odds = [ 1 : 2 : 113 ];
>> sum( odds )
ans =
        3249
>> % nth odd number = 2n - 1
>> [ 1 : 2 : 2*100 - 1 ]
ans =
  Columns 1 through 11
     1     3     5     7     9    11    13    15    17    19    21
  Columns 12 through 22
    23    25    27    29    31    33    35    37    39    41    43
  Columns 23 through 33
    45    47    49    51    53    55    57    59    61    63    65
  Columns 34 through 44
    67    69    71    73    75    77    79    81    83    85    87
  Columns 45 through 55
    89    91    93    95    97    99   101   103   105   107   109
  Columns 56 through 66
   111   113   115   117   119   121   123   125   127   129   131
  Columns 67 through 77
   133   135   137   139   141   143   145   147   149   151   153
  Columns 78 through 88
   155   157   159   161   163   165   167   169   171   173   175
  Columns 89 through 99
   177   179   181   183   185   187   189   191   193   195   197
  Column 100
   199
>> linspace( 1, 2*100-1, 100 )
ans =
  Columns 1 through 11
     1     3     5     7     9    11    13    15    17    19    21
  Columns 12 through 22
    23    25    27    29    31    33    35    37    39    41    43
  Columns 23 through 33
    45    47    49    51    53    55    57    59    61    63    65
  Columns 34 through 44
    67    69    71    73    75    77    79    81    83    85    87
  Columns 45 through 55
    89    91    93    95    97    99   101   103   105   107   109
  Columns 56 through 66
   111   113   115   117   119   121   123   125   127   129   131
  Columns 67 through 77
   133   135   137   139   141   143   145   147   149   151   153
  Columns 78 through 88
   155   157   159   161   163   165   167   169   171   173   175
  Columns 89 through 99
   177   179   181   183   185   187   189   191   193   195   197
  Column 100
   199
>> n = 2
n =
     2
>> sum( linspace( 1, 2*n-1, n ) )
ans =
     4
>> n = 3
n =
     3
>> sum( linspace( 1, 2*n-1, n ) )
ans =
     9
>> n = 4
n =
     4
>> sum( linspace( 1, 2*n-1, n ) )
ans =
    16
>> n = 2
n =
     2
>> sum( [1:n] )
ans =
     3
>> n = 3
n =
     3
>> sum( [1:n] )
ans =
     6

Notice any patterns to the sums?

• The sum of the first n odd naturals is n²
• The sum of the first n naturals is n(n+1)/2
• You’ll see both of these proved in courses such as Proofs

Next

Calculating with vectors

Read sections 4.6 - 4.7

Next Lecture