The Laplace Transform
Laplace transform is an operational tool for solving constant conceits linear differential equations.
The process of solution consists of three main steps:
? The given “hard" problem is transformed into a simple" equation.
? This simple equation is solved by purely algebraic manipulations.
? The solution of the simple equation is transformed back to obtain the solution of the given
problem.
Suppose that f is a real- or complex-valued function of the (time) variable t > 0 and (s) is a real
or complex parameter. We define the Laplace transform of f as:
The symbol ?? is the Laplace transformation, which acts on functions f = f (t) and generates a new
function, F(s) = L f (t) whenever the limit exists (as a finite number). When it does, the integral
is said to converge. If the limit does not exist, the integral is said to diverge and there is no Laplace
transform defined for f .
Examples: Find the Laplace transformations for the following functions:
Definition: Anm×n matrix is a rectangular array of numbers
(m rows and n columns) enclosed in brackets. The numbers
are called the elements of the matrix.
Examples:
(i) A 2 × 3 matrix has 2 rows and 3 columns:
A =
µ
1 2 3
5 6 7
¶
(ii) Here’s a 3 × 3 square matrix:
A =
0
@
1 2 3
5 6 7
8 9 10
1
A
(iii) Column vectors are matrices with only one column:
b =
0
@
1
5
8
1
A
(iv) Row vectors are matrices which only have one row:
b = (1 2 3) .
Unless specifically stated otherwise, we will assume that vectors
are column vectors.
1
Materials engineering Collage \\ Ceramic & construction materials department
Numerical Analysis \\Third stage
by \\ Dalya Hekmat
: Linear Algebra | Lecture 1 §1 Matrices and matrix algebra
A general real matrix, A 2 Rm×n with m×n elements is of
the form
A =
0
BBBBB@
a11 a12 a13 . . . a1n
a21 a22 a23 . . . a2n
a31 a32 a33 . . . a2n ...
...
...
. . . ...
am1 am2 am3 . . . amn
1
CCCCCA
(1)
We refer to the elements via double indices as follows
(i) The first index represents the row.
(ii) The second index represents the column.
Example
a32 is the element in row 3, column 2 of the matrix A.
Notation/Conventions:
Use lowercase boldface (or underlined) letters for vectors
a b c (or a, b, c)
Use uppercase boldface (or underlined) letters for matrices
A B C (or A, B, C)
Refer to the respective elements by lowercase letters with the
appropriate number of indices e.g.
المادة المعروضة اعلاه هي مدخل الى المحاضرة المرفوعة بواسطة استاذ(ة) المادة . وقد تبدو لك غير متكاملة . حيث يضع استاذ المادة في بعض الاحيان فقط الجزء الاول من المحاضرة من اجل الاطلاع على ما ستقوم بتحميله لاحقا . في نظام التعليم الالكتروني نوفر هذه الخدمة لكي نبقيك على اطلاع حول محتوى الملف الذي ستقوم بتحميله .
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