linear

Matrix Inverse

are fundamental concepts in linear algebra. They allow us to solve complex systems of equations efficiently.

matrix inverse notation

Properties of Matrix Inverses

Square Requirement

Inverse of Inverse

Inverse of Product

Transpose Relation

Square Requirement

Only square matrices (the same number of rows and columns) can have inverses.

Inverse of Inverse

Inverse of Product

Transpose Relation

Conditions for Matrix Invertibility

Square Matrix

Non-zero Determinant

Full Rank

Square Matrix

The matrix must have equal numbers of rows and columns. A non-square matrix cannot have an inverse.

Non-zero Determinant

The determinant must not equal zero. Matrices with zero determinants are called "singular" matrices.

Full Rank

The matrix must have full rank, meaning all rows and columns are linearly independent.

The Invertible Matrix Theorem

• Unique Solution

• Linear Independence

• Full Rank

• Non-zero Determinant

Unique Solution

Ax = b has a unique solution for every b.

Linear Independence

Columns form linearly independent set

Full Rank

Matrix has rank n

Non-zero Determinant

det(A) ≠ 0

LINEAR ALGEBRA

Is a branch of mathematics that deals with vectors, vector spaces,

and linear transformations.

Why is it IMPORTANT in COMPUTING

Graphic Rendering, Machine Learning, Cryptography, Search Engines, Data Compression

Graphic Rendering

Turning mathematical descriptions into 3D images in video games

and virtual reality.

Machine Learning

Training models to recognize patterns in massive datasets.

Cryptography

Encrypting and securing sensitive information

Search Engines

Optimizing how search results are ranked.

Data Compression

Reducing file sizes for storage and transmission

MATHEMATICAL FOUNDATION

Linear algebra is a branch of mathematics that

deals with vector spaces, linear transformations,

and systems of linear equations.

COMPUTATIONAL POWER

It provides a powerful framework for solving

problems in various fields, including computer

science, engineering, and physics.

Vectors

are fundamental

building blocks in linear

algebra, representing

quantities with both

magnitude and direction

Abstract Spaces

provide a

framework for working with

vectors and linear

transformations in a general

setting.

Linear Transformations

map vectors in one vector space to

vectors in another.

Matrix representation

provide a compact and efficient way to represent

and manipulate linear transformations.

Computer Graphics

used for transformations like rotation,

scaling, and translation in computer graphics.

Networking

Network analysis, data routing, and traffic flow

management rely on linear algebra techniques.

SCALAR

is a single number. It can be a real number, complex number, etc.

VECTOR

is an ordered list of numbers. It can

represent points in space, directions, or quantities

with both magnitude and direction.

MATRIX

is a rectangular array of numbers arranged

in rows and columns. Matrices are used to

represent linear transformations and systems of

linear equations.

Vectors

often represented as column

matrices.

Matrices

are

typically denoted by bold capital letters, with

elements identified by subscripts indicating

their row and column positions.

Robotics

Used to represent and control

the position and motion of

robots.

Computer Graphics

Used to manipulate images

and objects in computer

graphics

Machine Learning

Used to analyze and predict

data patterns in machine

learning.

Quantum Mechanics

Used to describe the behavior

of particles in quantum

mechanics.

Field

a set equipped with two operations,

addition (+) and multiplication (·), satisfying a set

of axioms. These axioms include closure,

associativity, commutativity, distributivity,

identity elements, and inverses.

Real Numbers (R)

The set of all _ numbers, including

rational and irrational numbers,

forms a field. They are used in

everyday calculations and are the

foundation for many mathematical

concepts.

Complex Numbers (C)

extends

the real numbers by including the

imaginary unit "i" (√-1). They are

essential in various areas of

mathematics, physics, and

engineering

Finite Fields

a finite number of elements. These fields are widely used in

coding theory, cryptography, and computer science.

Vector Addition

are closed under

vector addition, meaning that

the sum of any two vectors in

the space is also a vector in the

space.

Scalar Multiplication

allows

scaling vectors by a factor.

Multiplying a vector by a scalar

results in another vector within

the same space.

Vector Space Axioms

These operations must satisfy specific axioms, including associativity,

commutativity, identity elements, and inverses.

Subspaces

is a subset of a vector space that is closed

under vector addition and scalar multiplication.

Linear Independence

linearly independent if no vector in the

set can be expressed as a linear combination of the other

vectors in the set.

Basis

is a linearly independent set of

vectors that spans the entire space. Any vector in the space

can be expressed as a linear combination of the basis

vectors

Dimension

is the number of vectors in

a basis. It represents the number of independent directions

needed to span the entire space.

Matrix Multiplication

can be used to compose linear

transformations. Multiplying two matrices represents applying the

transformations sequentially

solving equations

provides methods for solving systems of linear equations, which are essential in many computational tasks

applications

Examples include finding solutions to optimization problems, modeling physical systems, and analyzing data.

| Substitution Method

| Rules

1. Solve ONE equation for ONE variable.

2. Substitute or Replace ONE expression into Equation.

3. Solve, then plug in the Equation.

| Elimination Method

| Rules

1. Add the equations to eliminate the y

If the y coefficients are different, look for their Least Common Multiple (LCM), and change the signs to make the sum result in 0.