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-5) Linear Dependence Coefficient It is a scalar number used to multiply a variable. Like 6z means 6 times z, and "z" is a variable, so 6 is a coefficient. Linear Combination If we take a set of matrices then multiply each of them by a scalar/coefficient and add all of them together then the resultant matrix is a linear combination of the set of matrices. Note that all the matrices involved in a linear combination need to have the same dimension (otherwise matrix addition would not be possible). Let's consider n number of matrices A 1 ,A 2 ,......,A n . all has same dimensions. Lets also consider n scalars/coefficients α 1 ,α 2 ,......,α n So, linear combination B will be B = α 1 A 1 + α 2 A 2 + ...... + α n A n Let's see few ways of calculation linear Combination / coefficients. Way 1 Holding the equation, B = α 1 A 1 + α 2 A 2 + ...... + α n A n . We conside...

LINEAR ALGEBRA (Part-4) Identity and Inverse Matrices Identity Matrix It is a square matrix that do not change any matrix or vector when multiplied with. Means If a matrix A 2,2 is multiplied by identity matrix I 2,2 then the result will be A 2,2 . So, A 2,2 ❌ I 2,2 = A 2,2 A 2,2 = [           I 2,2 = [           A 2,2 = [             A 2,2 = [         2 3   ❌       1 0   =       2❌1 ➕ 3❌0   2❌0 ➕ 3❌1   =       2 3     5 4           0 1           5❌1 ➕ 4❌0   5❌0 ➕ 4❌1           5 4   ]             ]             ]               ]       Identity matrix looks like above I 2,2 . Where dia...