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

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.