**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

**Matrix Addition**

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.