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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$\mathbf{X} = \left[\begin{array} {rrr} 1 & 0 \\ 2 & 5 \\ 3 & 1 \end{array}\right] $ + $ \left[\begin{array} {rrr} 4 & 0.5 \\ 2 & 5 \\ 0 & 1 \end{array}\right] $ = $\left[\begin{array} {rrr} 5 & 0.5 \\ 4 & 10 \\ 3 & 2 \end{array}\right] $
- you can only add matrices of the same dimension, and the result will be another matrix that's of the same dimension as the ones you just added.
$\mathbf{X} = \left[\begin{array} {rrr} 1 & 0 \\ 2 & 5 \\ 3 & 1 \end{array}\right] $ + $ \left[\begin{array} {rrr} 4 & 0.5 \\ 2 & 5 \end{array}\right] $ = Error
Quiz: What is
$\left[\begin{array} {rrr} 8 & 6 & 9 \\ 10 & 1 & 10 \end{array}\right] $ + $ \left[\begin{array} {rrr} 3 & 10 & 2 \\ 6 & 1 & -1 \end{array}\right] $?
$\left[\begin{array} {rrr} 11 & 16 & 11 \\ 16 & 2 & 9 \end{array}\right] $
Scalar (Real number) Multiplication
3 X $\left[\begin{array} {rrr} 1 & 0 \\ 2 & 5 \\ 3 & 1 \end{array}\right] $ = $\left[\begin{array} {rrr} 3 & 0 \\ 6 & 15 \\ 9 & 3 \end{array}\right] $ = $\left[\begin{array} {rrr} 1 & 0 \\ 2 & 5 \\ 3 & 1 \end{array}\right] $ X 3
$\left[\begin{array} {rrr} 4 & 0 \\ 6 & 3 \end{array}\right] $ / 4 = $\frac{1}{4}$ $\left[\begin{array} {rrr} 4 & 0 \\ 6 & 3 \end{array}\right] $ = $\left[\begin{array} {rrr} 1 & 0 \\ 1.5 & 0.75 \end{array}\right] $
Quiz: What is 2 X $\left[\begin{array} {rrr} 4 & 5 \\ 1 & 7 \end{array}\right] $?
$\left[\begin{array} {rrr} 8 & 10 \\ 2 & 14 \end{array}\right] $
Combination of Operands
3 X $\left[\begin{array} {rrr} 1 \\ 4 \\ 2 \end{array}\right] $ + $\left[\begin{array} {rrr} 0 \\ 0 \\ 5 \end{array}\right] $ - $\left[\begin{array} {rrr} 3 \\ 0 \\ 2 \end{array}\right] $ / 3
= $\left[\begin{array} {rrr} 3 \\ 12 \\ 6 \end{array}\right] $ + $\left[\begin{array} {rrr} 0 \\ 0 \\ 5 \end{array}\right] $ - $\left[\begin{array} {rrr} 1 \\ 0 \\ 0.66 \end{array}\right] $
= $\left[\begin{array} {rrr} 2 \\ 12 \\ 10.33 \end{array}\right] $
Quiz: what is $\left[\begin{array} {rrr} 4 \\ 6 \\ 7 \end{array}\right] $ / 2 - 3 X $\left[\begin{array} {rrr} 2 \\ 1 \\ 0 \end{array}\right] $?
$\left[\begin{array} {rrr} -4 \\ 0 \\ 3.5 \end{array}\right] $
Lecturer's Note
Addition and subtraction are element-wise, so you simply add or subtract each corresponding element:
$\left[\begin{array} {rrr} a & b \\ c & d \end{array}\right] $ + $\left[\begin{array} {rrr} w & x \\ y & z \end{array}\right] $ = $\left[\begin{array} {rrr} a+w & b+x \\ c+y & d+z \end{array}\right] $
Subtracing matrics:
$\left[\begin{array} {rrr} a & b \\ c & d \end{array}\right] $ - $\left[\begin{array} {rrr} w & x \\ y & z \end{array}\right] $ = $\left[\begin{array} {rrr} a-w & b-x \\ c-y & d-z \end{array}\right] $
To add or subtract two matrices, their dimensions must be the same.
In scalar multiplication, we simply multiply every element by the scalar value:
$\left[\begin{array} {rrr} a & b \\ c & d \end{array}\right] $ * x= $\left[\begin{array} {rrr} ax & bx \\ cx & dx \end{array}\right] $
In scalar division, we simply divide every element by the scalar value:
$\left[\begin{array} {rrr} a & b \\ c & d \end{array}\right] $ * x= $\left[\begin{array} {rrr} a/x & b/x \\ c/x & d/x \end{array}\right] $
Experiment below with the Octave/Matlab commands for matrix addition and scalar multiplication. Feel free to try out different commands. Try to write out your answers for each command before running the cell below.
% Initialize matrix A and B
A = [1, 2, 4; 5, 3, 2]
B = [1, 3, 4; 1, 1, 1]
% Initialize constant s
s = 2
% See how element-wise addition works
add_AB = A + B
% See how element-wise subtraction works
sub_AB = A - B
% See how scalar multiplication works
mult_As = A * s
% Divide A by s
div_As = A / s
% What happens if we have a Matrix + scalar?
add_As = A + s
