This is a brief summary of ML course provided by Andrew Ng and Stanford in Coursera.
You can find the lecture video and additional materials in
https://www.coursera.org/learn/machine-learning/home/welcome
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Matrix: Rectangular array of numbers:
$\mathbf{X} = \left[\begin{array} {rrr} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{array}\right] $
$\mathbf{X} = \left[\begin{array} {rrr} 1401 & 191 \\ 1371 & 821 \\ 949 & 1437 \\ 147 & 1448 \end{array}\right] $
Dimension of matrix: number of rows x number of columns
3X3 Matrix $(R^{3x3})$ and 4X2 $(R^{4x2})$ Matrix above.
Quiz: Which of the following statements are true? Check all that apply.
1. $\mathbf{X} = \left[\begin{array} {rrr} 1 & 2 \\ 4 & 0 \\ 0 & 1 \end{array}\right] $ is a 3 X 2 matrix.
2. $\mathbf{X} = \left[\begin{array} {rrr} 0 & 1 & 4 & 2 \\ 3 & 4 & 0 & 9 \end{array}\right] $ is a 4 X 2 matrix.
3. $\mathbf{X} = \left[\begin{array} {rrr} 0 & 4 & 2 \\ 3 & 4 & 9 \\ 5 & -1 & 0 \end{array}\right] $ is a 3 X 3 matrix.
4. $\mathbf{X} = \left[\begin{array} {rrr} 1 & 2 \end{array}\right] $ is a 1 X 2 matrix.
Answer: 1, 3, 4
Matrix Elements (entries of matrix)
$\mathbf{A} = \left[\begin{array} {rrr} 1401 & 191 \\ 1371 & 821 \\ 949 & 1437 \\ 147 & 1448 \end{array}\right] $
$A_{ij}$= "i,j entry" in the $i^{th}$ row, $j^{th}$ column.
$A_{11}$ = 1402,
$A_{12}$ = 191,
$A_{32}$ = 1437,
$A_{41}$ = 147,
$A_{43}$ = undefined (error)
Quiz: Let A be a matrix shown below. $A_32$ is one of the elements of this matrix.
$\mathbf{A} = \left[\begin{array} {rrr} 85 & 76 & 66 & 5 \\ 94 & 75 & 18 & 28 \\ 68 & 40 & 71 & 5 \end{array}\right] $
What is the value of $A_{32}$ ?
Answer: 40
Vector: An n X 1 matrix (1 column)
$\mathbf{X} = \left[\begin{array} {rrr} 460 \\ 232 \\ 315 \\ 178 \end{array}\right] $ -> $R^4$
This is 4-dimensional vector.
$y_i = i^{th}$ element
$y_1$ = 460, $y_2$ = 232, $y_3$ = 315
Othen than explicitly specified, use 1-indexed vector.
And use capital letter for matrices, and lower case to refer to either numbers, scalar, or vectors.
Lecturer's Note
Matrices are 2-dimensional arrays:
$\mathbf{A} = \left[\begin{array} {rrr} a & b & c \\ d & e & f \\ g & h & i \\ j & k & l \end{array}\right] $
The above matrix has four rows and three columns, so it is a 4 x 3 matrix.
A vector is a matrix with one column and many rows:
$\mathbf{A} = \left[\begin{array} {rrr} w \\ x \\ y \\ z \end{array}\right] $
So vectors are a subset of matrices. The above vector is a 4 x 1 matrix.
Notation and terms:
- $A_{ij}$ refers to the element in the ith row and jth column of matrix A.
- A vector with 'n' rows is referred to as an 'n'-dimensional vector.
- $v_i$ refers to the element in the ith row of the vector.
- In general, all our vectors and matrices will be 1-indexed. Note that for some programming languages, the arrays are 0-indexed.
- Matrices are usually denoted by uppercase names while vectors are lowercase.
- "Scalar" means that an object is a single value, not a vector or matrix.
- R refers to the set of scalar real numbers.
- $R^{n}$ refers to the set of n-dimensional vectors of real numbers.
Run the cell below to get familiar with the commands in Octave/Matlab. Feel free to create matrices and vectors and try out different things.
% The ; denotes we are going back to a new row.
A = [1, 2, 3; 4, 5, 6; 7, 8, 9; 10, 11, 12]
% Initialize a vector
v = [1;2;3]
% Get the dimension of the matrix A where m = rows and n = columns
[m,n] = size(A)
% You could also store it this way
dim_A = size(A)
% Get the dimension of the vector v
dim_v = size(v)
% Now let's index into the 2nd row 3rd column of matrix A
A_23 = A(2,3)
A = 1 2 3 4 5 6 7 8 9 10 11 12 v = 1 2 3 m = 4 n = 3 dim_A = 4 3 dim_v = 3 1 A_23 = 6
