Solving Matrix-Vector Equations Eric Eager Data Scientist at Pro - - PowerPoint PPT Presentation

▶
solving matrix vector equations
SMART_READER_LITE
LIVE PREVIEW

Solving Matrix-Vector Equations Eric Eager Data Scientist at Pro - - PowerPoint PPT Presentation

Dec 22, 2023 210 likes •598 views

DataCamp Linear Algebra for Data Science in R LINEAR ALGEBRA FOR DATA SCIENCE IN R Solving Matrix-Vector Equations Eric Eager Data Scientist at Pro Football Focus DataCamp Linear Algebra for Data Science in R Motivation - Can These Vectors

slide-1
SLIDE 1

DataCamp Linear Algebra for Data Science in R

Solving Matrix-Vector Equations

LINEAR ALGEBRA FOR DATA SCIENCE IN R

Eric Eager

Data Scientist at Pro Football Focus

slide-2
SLIDE 2

DataCamp Linear Algebra for Data Science in R

Motivation - Can These Vectors Make That Vector?

slide-3
SLIDE 3

DataCamp Linear Algebra for Data Science in R

Motivation - Can These Vectors Make That Vector?

slide-4
SLIDE 4

DataCamp Linear Algebra for Data Science in R

Motivation - Can These Vectors Make That Vector?

slide-5
SLIDE 5

DataCamp Linear Algebra for Data Science in R

Motivation - Can These Vectors Make That Vector?

> A%*%x [,1] [1,] -1 [2,] 4 [3,] 0 > A[, 1]*x[1] + A[,2]*x[2] [1] -1 4 0

slide-6
SLIDE 6

DataCamp Linear Algebra for Data Science in R

Motivation - Can These Vectors Make That Vector?

slide-7
SLIDE 7

DataCamp Linear Algebra for Data Science in R

Example of a Matrix-Vector Equation

slide-8
SLIDE 8

DataCamp Linear Algebra for Data Science in R

Example of a Matrix-Vector Equation

slide-9
SLIDE 9

DataCamp Linear Algebra for Data Science in R

Example of a Matrix-Vector Equation

slide-10
SLIDE 10

DataCamp Linear Algebra for Data Science in R

Let's practice!

LINEAR ALGEBRA FOR DATA SCIENCE IN R

slide-11
SLIDE 11

DataCamp Linear Algebra for Data Science in R

Matrix-Vector Equations - Some Theory

LINEAR ALGEBRA FOR DATA SCIENCE IN R

Eric Eager

Data Scientist at Pro Football Focus

slide-12
SLIDE 12

DataCamp Linear Algebra for Data Science in R

A Matrix-Vector Equation Without a Solution

Inconsistent

slide-13
SLIDE 13

DataCamp Linear Algebra for Data Science in R

A Matrix-Vector Equation with Infinitely-Many Solutions

Consistent (but infinitely-many solutions)

slide-14
SLIDE 14

DataCamp Linear Algebra for Data Science in R

A Matrix-Vector Equation with a Unique Solution

Consistent (unique solution)

slide-15
SLIDE 15

DataCamp Linear Algebra for Data Science in R

Properties of Solutions to Matrix-Vector Equations - Exactly One Solution

slide-16
SLIDE 16

DataCamp Linear Algebra for Data Science in R

Properties of Solutions to Matrix-Vector Equations - No Solutions

slide-17
SLIDE 17

DataCamp Linear Algebra for Data Science in R

Properties of Solutions to Matrix-Vector Equations - Infinitely- Many Solutions

slide-18
SLIDE 18

DataCamp Linear Algebra for Data Science in R

Properties to Ensure A Unique Solution to A = x⃗ b⃗

If A is an n by n square matrix, then the following conditions are equivalent and imply a unique solution to A = : The matrix A has an inverse (is invertible) The determinant of A is nonzero The rows and columns of A form a basis for the set of all vectors with n elements x⃗ b⃗

slide-19
SLIDE 19

DataCamp Linear Algebra for Data Science in R

Properties to Ensure A Unique Solution to A = x⃗ b⃗

Computing the Inverse of A (if it Exists) Computing the Determinant of A

A [,1] [,2] [1,] 1 -2 [2,] 0 4 solve(A) [,1] [,2] [1,] 1 0.50 [2,] 0 0.25 det(A) [1] 4

slide-20
SLIDE 20

DataCamp Linear Algebra for Data Science in R

Let's practice!

LINEAR ALGEBRA FOR DATA SCIENCE IN R

slide-21
SLIDE 21

DataCamp Linear Algebra for Data Science in R

Solving Matrix-Vector Equations

LINEAR ALGEBRA FOR DATA SCIENCE IN R

Eric Eager

Data Scientist at Pro Football Focus

slide-22
SLIDE 22

DataCamp Linear Algebra for Data Science in R

Solving Matrix-Vector Equations

slide-23
SLIDE 23

DataCamp Linear Algebra for Data Science in R

Solving Matrix-Vector Equations

slide-24
SLIDE 24

DataCamp Linear Algebra for Data Science in R

Solving Matrix-Vector Equations

Solving A = using = A :

A [,1] [,2] [1,] 1 -2 [2,] 0 4 b [1] 1 -2

x⃗ b⃗ x⃗

−1b⃗

x <- solve(A)%*%b print(x) [,1] [1,] 0.0 [2,] -0.5

slide-25
SLIDE 25

DataCamp Linear Algebra for Data Science in R

Solving Matrix-Vector Equations

Checking your solution by plugging in the solution : Which is equal to the given :

x <- solve(A)%*%b print(x) [,1] [1,] 0.0 [2,] -0.5

x⃗

A%*%x [,1] [1,] 1 [2,] -2

b⃗

print(b) [1] 1 -2

slide-26
SLIDE 26

DataCamp Linear Algebra for Data Science in R

Additional Conditions for Unique Solutions

Thus, the only solution to the homogeneous equation A = is the trivial solution = . x⃗ 0⃗ x⃗ 0⃗

slide-27
SLIDE 27

DataCamp Linear Algebra for Data Science in R

Additional Conditions for Unique Solutions

A [,1] [,2] [1,] 1 -2 [2,] 0 4 b <- rep(0, 2) print(b) [1] 0 0 > solve(A)%*%b [,1] [1,] 0 [2,] 0

slide-28
SLIDE 28

DataCamp Linear Algebra for Data Science in R

Conditions for a Unique Solution to Matrix-Vector Equations

If A is an n by n square matrix, then the following conditions are equivalent and imply a unique solution to A = : The matrix A has an inverse (is invertible) The determinant of A is nonzero The rows and columns of A form a basis for the set of all vectors with n elements The homogeneous equation A = has just the trivial ( = 0) solution x⃗ b⃗ x⃗ 0⃗ x⃗

slide-29
SLIDE 29

DataCamp Linear Algebra for Data Science in R

Let's Practice!

LINEAR ALGEBRA FOR DATA SCIENCE IN R

slide-30
SLIDE 30

DataCamp Linear Algebra for Data Science in R

Other Considerations for Matrix-Vector Equations

LINEAR ALGEBRA FOR DATA SCIENCE IN R

Eric Eager

Data Scientist at Pro Football Focus

slide-31
SLIDE 31

DataCamp Linear Algebra for Data Science in R

More Equations than Unknowns

slide-32
SLIDE 32

DataCamp Linear Algebra for Data Science in R

More Equations than Unknowns

slide-33
SLIDE 33

DataCamp Linear Algebra for Data Science in R

Fewer Equations than Unknowns

slide-34
SLIDE 34

DataCamp Linear Algebra for Data Science in R

Some Options for Non-Square Matrices

Row Reduction (By Hand, Difficult for Big Problems) Least Squares (If More Rows Than Columns - Used in Linear Regression) Singular Value Decomposition (If More Columns Than Rows - Used in Principal Component Analysis) Generalized or Pseudo-Inverse

slide-35
SLIDE 35

DataCamp Linear Algebra for Data Science in R

Moore-Penrose Generalized Inverse

library(MASS) A [,1] [,2] [1,] 2 3 [2,] -1 4 [3,] 1 7 ginv(A) [,1] [,2] [,3] [1,] 0.3333333 -0.30303030 0.03030303 [2,] 0.0000000 0.09090909 0.09090909 ginv(A)%*%A [,1] [,2] [1,] 1 -1.110223e-16 [2,] 0 1.000000e+00 A%*%ginv(A) [,1] [,2] [,3] [1,] 0.6666667 -0.3333333 0.3333333 [2,] -0.3333333 0.6666667 0.3333333 [3,] 0.3333333 0.3333333 0.6666667

slide-36
SLIDE 36

DataCamp Linear Algebra for Data Science in R

Moore-Penrose Generalized Inverse

{r} <- ginv(A)%*%b A%*%x [,1] [1,] 1 [2,] 7 [3,] 8 {{2}}

A [,1] [,2] [1,] 2 3 [2,] -1 4 [3,] 1 7 b [1] 1 7 8

slide-37
SLIDE 37

DataCamp Linear Algebra for Data Science in R

Let's Practice

LINEAR ALGEBRA FOR DATA SCIENCE IN R