Equality of Vectors:

Definition 1:

Vectors V=(a,b,c,...,z) and W=(1,2,3,4,5,...,n) are said to be equivalent if a=1 ; b=2 ; c=3 ; ...

We indicate the equality by writing V=W

Definition 2:

V+W= (a+1, b+2, ...,Z+n)

KV= (Ka, Kb, Kc, ..., Kn)

-V=(-a,-b,-c,...,-z)

Notification:

As far as we know there are two notations for vectors:

The comme delimited form: V=(a,b,c,...,z) also called the row vector form

The column vector form

Will start this cours by an Introduction to matrices:

Elementary row operations:

- Multiply an equation through by a nonzero constant
- Interchange two equations
- Add a multiple of one equation to another

Echelon Forms:

Before getting started with Jordan and Gaussian you have first to learn some properties about echelon forms:

A matrix is in Row Echelon Form if:

- The zero rows are grouped together at the bottom of the matrix.
- In any two successive rows, the leading entry of the second one should be on the right of the leading entry of the first one.
- In a non zero row the leading entry is always one.

A matrix is in Reduced Row Echelon Form if:

- It is in Row Echelon form.
- Eache column that contain a leading entry 1 has zeros everywhere.

Notification:

Every matrix has a unique reduced row echelon form.

Every matrix has an infinite number of row echelon forms.

Augmented Matrices and Jordan-Gausse and Gaussian elimination:

Solving Linear Systems by Row Reduction:

Matrix Operations:

The Trace:

If A is a square matrix, then the trace of A, denoted by tr(A), is defined to be the sum of the entries on the main diagonal of A.

Zero matrix VS Identity matrix

- A is a zero matrix if and only if all its entries are equal to ZERO
- A square matrix with 1's on the main diagonal and zeroq elswhere is called an identity matrix.

Inverse of Matrix: Theory

Inverse of a 2 by 2 Matrix:

The inverse of a matrix using row operations.

here it's a 3 by 3 but it workes for all the sizes

Properties of Diagonal Matrix and Transpose:

Elementary matrices; A method for finding the inverse of the matrix A

Solving Linear Systems by matrix inversion:

Subspace

Spaning and Basis

Some practice:

This is it.The End

This is almost all about Mat 120, we hope that you enjoyed our company and that you took full advantage of what we presented to you.

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