NOTES

# 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]])

A

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]])

A

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)

r

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