Applied Linear algebra

Row Operations

The 3 elementary row operations:

1. Swap two rows.

2. Multiply a row by a nonzero scalar (e.g., divide a row by 3).

3. Add or subtract a multiple of one row to/from another.

Gaussian Elimination

Pivot Column

Pivot – first none zero in a row of matrix

Row enchelon form

Row reduced enchelon form

Basically the difference between this and other, is the leading entry 1 has to be the only none zero in its columnt.first none zero in each row has to be 1

Pivot position and pivot column (Has to be be in row encholen form)

Basic and free variables

Basic variable – if it corresponds to a pivot column

Free variable -  if does Not correspond to a pivot column

Consisent system

·         A consistent matrix is an augmented matrix that represents a system of equations with at least one solution.

·         Unique no Free variables, no contradictions

Rank

The number of pivot columns (i.e., the number of leading 1s) in its row echelon form

Homogeneous Equations

where all constant are 0.

Homogeneous has at least one solution.

• If rank <n, then there are infinite solutions.

• If rank =n, then the only solution is the trivial one.

(n is the number of columns)

Trivial vs non trivial

Trivial if everything is 0.

Non trivial not 0.

Non trivial if there are any free variables

Linear Combinations and Basic Solutions

Matrix Addition

Has to be same size to add and minus

Scalar multiplication matrices

Vector Matrix Multiplication

Multiplying matrices

Transpose

Symmetric and skew symmetric

Identity matrix

Invertible matrix

Invertible matrices are basically matrices that have an inverse

Finding inverse of a matrix

A matrix with no inverse cannot get to this form (has to be reduced row enncholon form)

Solving linear system with matrices

Ax = B

(A^-1)Ax = (A^-1)B

x = (A^-1)B

Elementary matrices

Inverse of elementary matrices

·         Every elementary matrices is invertible

·         An elementary matrix represents a single row operation done to an identity matrix. Its inverse is simply the matrix that undoes that operation.

Triangular matrices

·         A triangular matrix is a square matrix (same number of rows and columns) where all entries on one side of the main diagonal are zero.

Determinant

single number calulated from a square matrix (2×2)(3×3)(….)

Matrix minros

·         Basically doing the determinate at that point we are looking at for minors

Cofactors

Determinant cofactor expansion

(can choose a row or column)

Determinant triangular matrix

Properties of determinants

Determinant usign row operations

·         Make matrix into a triangular matrix

Determinant cursory inspection

·         Cursory inspection means evaluating a determinant quickly by observing special patterns or properties — without full cofactor expansion or row reduction.

·         It saves time, especially on multiple choice tests or when recognizing zero determinants.

Vector formula

Position Vector

Properteis of Vector

Vector Linear combination

Distance between points

Length and Unit vector

Dot product

if dot product 0 means hey are perpendicular(othrogonal)( right angle) to each other

Projection vector

w1, and w2 are components of Vector U

w1 is projection of vector U onto vector V(thats how its described)(component of Vector U that is parralel to vector V

w2 is teh component of vector U that is othorgonal to vector V

Projection tells you how far one vector goes in the direction of another vector.

Why we use projection: 1. Shortest distance to a line or plane

Want to know the closest point on a line to a point? Use projection.
→ You find where the "shadow" of the point lands on the line.

Parallel and perpendicular vectors

Paremetic lines

Plane Normal vector and Vector equation

so if we get 0 then it means that P is a point on the plane

Scalar equation of a plane in R3

Find equation of a plane

Span of vectors

Describe vectors minum span

Solvign linear indpenetn

Vectors in a span

if linearly indpenedent then spans all of

Linearly Dependent

ANY COLUMN LACKS pivot then its a linearly Dependet

Linearnily Indpendent

Rule for linearly indpendent or Dependent on span

Subspace

Subspace are spans

Dimention of a subspace

Findind dimension of Subspace and basis

Solving subspace

Solvign subspace 2 (remmerb is a supspace then linearly indpendent)

Rank of a matrix

Row Space

Any row that has a pivot colum is part of teh row space

Column space

Any column that has a pivot column is part of column space

null space or kernal space

what multplied by A gives us 0

Rules of row space

rules for column space

Orthogonal set

Orothornamal set

Fourier expansion

Orothogonal matrixes

Orthornamal basis

Grand smit processs

Finding orhtornomal basis

Linear Transformation

Transformatin not linear

Linear trnasformation and projection

Matrix of linear transformation

Properteis of Linear Transformation

Composite of linear Transformation

One to one Linear Transform

Onto Linear Transform

Isophomersim

Properties of isophomerism

if det(A) note equal 0 isopherims (probaby dont use this)

Isopomerism subspace

matrix Isophermerism

Isophermism more with images

Prove matrix is invetitable

Inverse 2 by 2 matrix

Some more kernal space and others

More practice kker basia

rememeber teh reson its one and all (x+y) is the first rwo and (1 1 0)

all possible to be orthogonal

Orthogonal projection in liek span

Orthogoanl complement

Poitn in a plane closest to a given point

Least square solution

Least square soltuions more

solve some system for fun