Linear Algebra Basics

Useful terms and ideas

diagonal matrix: Matrix with zero values except on the diagonal.
change of basis matrix: Taking the product of this matrix provides a new basis, which is helpful to us if that basis transformation sets us up for an operation that requires the matrix to be in a particular form. For instance, a change of basis matrix could diagonalize the matrix, such as the choice of the orthogonal matrix as the change of basis matrix in PCA.
eigen values: These are the lambda values that satisfy the following equation where A is our input data, and x is the eigenvector: $A x = \lambda x$.
eigen vectors (characteristic vector): In a linear transformation, $L(x)$, then $x$ is an eigenvalue of $L()$ if $L(x)$ is a scalar multiple of $x$.
matrix transformations include reflection, orthogonal projection, rotation, compression, and shearing.
Because they are so useful, you can choose your eigenvectors as your basis (eigenbasis), provided they span your space
In 1751, Leonhard Euler proved that any body has a principal axis of rotation.

Scipy example

We are entering three pies into the bakeoff, and want to make sure we win. We’ve tried a lot of different ingredients, but ultimatley have found that taste comes down to butter, salt, sugar, and grandma’s secret spice blend.

import scipy.linalg as la
import numpy as np

w = 1 #tbl_of_butter
x = 1 #dash_of_salt
y = 1 #spoonful_of_sugar
z = 1 #undisclosed_measurement_of_spice

pie1 = 3*w + 5*x + 30*y + 10*z
pie2 = 2*w + 3*x + 40*y + 15*z
pie3 = 4*w + 2*x + 10*y + 30*z
pie4 = 10*w + 2*x + 10*y + 30*z

A = np.array([[3,5,30,10],[2,3,40,15],[4,2,10,30], [10,2,10,30]])

array([[ 3,  5, 30, 10],
       [ 2,  3, 40, 15],
       [ 4,  2, 10, 30],
       [10,  2, 10, 30]])

inv = la.inv(A)

array([[ 4.20539024e-19,  3.48136945e-18, -1.66666667e-01,
         1.66666667e-01],
       [ 3.55932203e-01, -2.71186441e-01,  1.15819209e-01,
        -9.88700565e-02],
       [-2.03389831e-02,  4.40677966e-02, -2.09039548e-02,
         5.64971751e-03],
       [-1.69491525e-02,  3.38983051e-03,  5.48022599e-02,
        -1.75141243e-02]])

TEST

A = np.array([[2, 3, -1], [1, -1, 0], [0, 5, 2], [3, 2, 1]])

array([[ 2,  3, -1],
       [ 1, -1,  0],
       [ 0,  5,  2],
       [ 3,  2,  1]])

test1 = np.array([2, 3, -7, 3])
test2 = np.array([0, 0, 0, 0])
test3 = np.array([1, 1, 1, 1])

x1 = la.lstsq(A, test1)

orth = la.orth(A)

r = np.ptp(A,axis=1)

array([4, 2, 5, 2])