বুঝাও (Teach Me)

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 consider n=2, α 1 =2, α 2 =-1 and

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 diagonally all values are 1 and rest are 0. Identity matrix is a square matrix so its row and column are always same. Like I 2,2 , I 4,4 , I 6,6 , I 3,3 etc. Matrix Inverse Matrix inverse is a bit complex. Don't worry we will not use this a lot. But we need it to verify some property. We will not go deep into inverse matrix. You can learn that in source (

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    .       5   =       2❌4 ➕ 1❌5   =       13     3  2          ]             3❌4 ➕ 2❌5           22    ]                         ]           ]   Remember the reverse will also hold. means consider 1X2 vector b and

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

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