LINEAR ALGEBRA (Part-6) - Norms

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LINEAR ALGEBRA (Part-6)

Norms

Norms

Norm is used to measure the size/length/magnitude/distance of a vector.
Like for example we know the Normal distance between two points x1=2 and x2=4 is x2-x1=4-2=2
Another example is the Euclidean distance between two points x1=2 and x2=4 is √(4-2)2=2
In general Norm is written by LP norm = ||x||p = (∑|xi|p)1/p Where p = 1 or 2

L1 Norm

The L1 norm is calculated as the sum of the absolute vector values.
Let's put 1 in the above general norm equation. Here p=1 as L1 Norm.
||x||p = (∑|xi|p)1/p
||x||1 = (∑|xi|1)1/1
||x||1 = ∑|xi|
So it is just the sum of the absolute vector values.
consider vector v = [1 -2 6]
||v||1 = |1| + |-2| + |6| = 1+2+6 = 9

L2 Norm

It is also called Euclidean norm.
The L2 norm is calculated as root over of the sum of the squared vector values.
Let's put 2 in the above general norm equation. Here p=2 as L2 Norm.
||x||p = (∑|xi|p)1/p
||x||2 = (∑|xi|2)1/2
1/2 is equal to root over "√".
||x||2 = √∑|xi|2
So it is just root over of the sum of the squared vector values.
consider vector v = [1 -2 6]
||v||2 = √ ( 12 + (-2)2 + 62 ) = √(1+4+36) = √41 = 6.403
L2 Norm is used so frequently that it is normally written as ||x||
L2 Norm can also be calculated by √xxT
√xxT = √ ( [1 -2 6] . [1 -2 6]T ) = √ ( (1X1) + (-2X-2) + (6X6) )
√xxT = √(1+4+36) = √41 = 6.403

Max Norm

It is represented as L∞
The max norm is calculated as returning the maximum value of the vector.
consider vector v = [1 -2 6] Then, L∞ = max(v) = 6

Frobenius norm

It is the same as L2 Norm.

SOURCE

1) Deep Learning (Ian Goodfellow, Yoshua Bengio, Aaron Courville)

2) https://machinelearningmastery.com/vector-norms-machine-learning

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LINEAR ALGEBRA (Part-2) - Multiplying Matrices

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LINEAR ALGEBRA (Part-2) Multiplying Matrices Standard Calculation We can add, subtract, divide or multiply a matrix with a scalar value. These operations will be element wise. Let's consider a 3✖2 Matrix A 3,2 . See below. A 3,2 = [                                     1 2                                     2 3                                     3 1                                     ]                                     Let's add a scalar value 5 with it. A 3,2 = [ A 3,2 = [  A 3,2 = [ 1 2 5✚1  5✚2  6 7 5 ✚ 2 3 =  5✚2 5✚3  = 7 8 3 1 5✚3  5✚1  8 6 ] ]  ] Let's subtract a scalar value 5 with it. A 3,2 = [ A 3,2 = [  A 3,2 = [ 1 2 5➖1  5➖2  4 3 5 ➖ 2 3 =  5➖2 5➖3  = 3 2 3 1 5➖3  5➖1  2 4 ] ]  ] Let's multiply a scalar value 5 with it. A 3,2 = [ A 3,2 = [  A 3,2 = [ 1 2 5❌1  5❌2  5 10 5 ❌ 2 3 = 5❌2 5❌3  = 10 15 3 1 5❌3  5❌1  15 5 ] ]  ] Let's divide a scalar value 5 with it. A 3,2 = [ A 3,2
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What Is derivative? (Part - 1)

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What Is derivative? (Part - 1) Derivative is a simple thing. It’s just the difference of a variable. But how? Think of time, let a person started a journey towards bus stand at time t1 = 3 and reached at t2 = 7. In this case the derivative dt = t2-t1 = 7-3 = 4. Easy! But in real life things are not that easy. Because we have do derive the difference of a variable with respect to another variable unlike our time example where there was only one variable. For example we can see the equation y = x. Here we have to find the difference in y with respect to the change in variable x. In other word we have to find the derivative of y with respect to x which is dy/dx. Fortunately for us this is very easy. Pretty easy right. But we are going to prove it so that we can use the same theory to prove more complex derivatives. So we are going to plot this function into graph. Let’s find some value of x and values of y for the respective value of x below.
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LINEAR ALGEBRA (Part-1) - Scalars, Vectors, Matrices and Tensors

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LINEAR ALGEBRA (Part-1) Scalars, Vectors, Matrices and Tensors Scalars Scalars are a single number. like 5 or 8 or 9 etc. you can say age=23 here 23 is a scalar value. Vectors Vectors are a series/array of elements. They are represented between a square bracket with spaces. Like below. ages = [    23    25    33    54 ] Here, [    23    25    33    54 ] is a Column Vector of ages. The Row Vector of ages will look like ages = [23 25 33 54] each element of a vector can be represented by a subscript number like ages 1 =23 or ages 3 =33 etc. the numbers of a vector do not have to be of same thing like. box = [    2    4    8    "wood" ] here the box vector represents, box 1 = height = 2 inch box 2 = width = 4 inch box 3 = weight = 8 gram box 4 = made of = wood Matrix Matrix is a 2D array of elements. Like below. box = [    2    3    2    4    4    4    5    2    8    12    5
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