বুঝাও (Teach Me)

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 x 1 =2 and x 2 =4 is x 2 -x 1 =4-2=2 Another example is the Euclidean distance between two points x 1 =2 and x 2 =4 is √(4-2) 2 =2 In general Norm is written by L P norm = ||x|| p = (∑|x i | p ) 1/p Where p = 1 or 2 L 1 Norm The L 1 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 L 1 Norm. ||x|| p = (∑|x i | p ) 1/p ||x|| 1 = (∑|x i | 1 ) 1/1 ||x|| 1 = ∑|x i | 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 L 2 Norm It is also called Euclidean norm. The L 2 norm is calculated as root over of the sum of the squared vector values. Let's put 2 in the above general norm equation....

LINEAR ALGEBRA (Part-3) Useful Properties Information Below, we will see some properties of Matrix. Remember there are many more then listed. Property 1 A(B + C) = AB + AC Property 2 A(BC) = (AB)C Property 3 A(BC) = (AB)C Property 4 AB ≠ BA For some cases AB = BA is true. But not for all. Property 5 A T B = B T A Property 6 (AB) T = B T A T Property 7 A T B = (A T B) T = B T A Property 8 A ❌ b = c The multiplication of a matrix and a vector results in a vector. Let's consider 3X2 matrix A and 2X1 vector b. So A ❌ b will result in 3X1 vector c. see below A 3,2 =  [           b 2,1 = [         c 3,1 = [         c 3,1 = [       1  3            4           1❌4 ➕ 3❌5           19     2  1    .     ...

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                                     ]          ...