# Linear Algebra

Introduction

1. It is a branch of mathematics
2. Concerned with the study of matrices, vectors, vector spaces (also called linear spaces), linear maps (also called linear transformations), and systems of linear equations.
3. Type “help matfun” (Matrix functions – numerical linear algebra) for more information or type “help elmat” (Elementary matrices and matrix manipulation).

Functions for Linear Algebra

Vectors

1. Vector x is given as:

2. “Transpose” of vector x is given as:
x’= [ x1 x2 x3 — xn ] ϵ R1xn

3. The “Length” of vector x:
||x||= √x’x = √(x1^2+x2^2+..+xn^2)
4. “Orthogonality” is given as:
x’y=0

Matrices

1. Matrix A is given as:

2. “Transpose” of matrix A is given as:

3. The Diagonal elements of matrix A is the vector:

4. The Diagonal matrix ⋀ is given by:

5. The Identity matrix I is given as:

Matrix Multiplication

Given matrices A ϵ Rnxm and B ϵ Rmxp , then
C= AB ϵ Rnxp

Note that:
AB = BA
A(BC) = (AB)C
(A+B)C = AC+BC
C(A+B) = CA+CB

Given matrices A ϵ Rnxm and B ϵ Rnxm , then
C= A+B ϵ Rnxm

Determinant

Given a matrix A ϵ Rnxn , then
det(A) =|A|
Given 2×2 matrix:

Then det(A) = |A| = a11 a22 – a21 a12

Note that:
det (AB) = det(A) det(B)
and
det(A’) = det(A)

Inverse Matrices

Inverse Matrices: The inverse of quadratic matrix A ϵ Rnxn is defined by:
A^-1
if
A^-1 x A = A x A^-1 = I
For 2×2 matrix

Then A^-1 is given by

Eigenvalues

Given A ϵ Rnxn , then eigenvalues is defined as:
det(lamda x I – A) = 0

Solving Linear Equations

Given following equations:
x1 + 2×2 = 5
3×1 + 4×2 = 6
7×1 + 8×2 = 9
To find the solution, backlash operator “\” is used.