Forms of Linear Equations Questions and Answers

Recent questions in Forms of linear equations
Brice Colon 2022-05-18 Answered

Complex form of Fourier's Theorem
Any periodic function can be decomposed into a linear combination of complex exponentials.
Proof
Consider a complex exponential with period T 0 :
e i 2 π T 0 t
Hence, we have:
0 T 0 e i 2 π T 0 t d t = 0 ( 1 )
Let us claim that for a periodic function x ( t ) we can write:
x ( t ) k = N N C k e i 2 π k T 0 t ,
where 2 N + 1 is the number of frequency components used. As N , we have:
x ( t ) = k = C k e i 2 π k T 0 t ( 2 )
Consider:
v k ( t ) := e i 2 π k T 0 t ( 3 )
Then we have:
v k ( t + T 0 ) = v k ( t )
Hence, v k ( t ) is a periodic function. Now,
0 T 0 v k ( t ) v l ( t ) d t
= 0 T 0 e i 2 π ( k l ) T 0 t d t
= 0 ( k l )
T 0 ( k = l ) ( 4 )
Thus, v k ( t ) is orthogonal. Moreover, if we assume equation (2) to be valid , then we can multiply both its sides by v l ( t ) and integrate over T 0 we get:
C k = 1 T 0 0 T 0 x ( t ) v k ( t ) d t ( 5 )
Putting this value in equation (2) we get back x ( t ). Hence, any periodic function can be decomposed into a linear combination of complex exponentials.
Real form of Fourier's Theorem
An arbitrary periodic function F ( t ) with period T can be decomposed into a linear combination of the functions f n ( t ) and g n ( t ) where,
f n ( t ) = sin 2 π n t T
g n ( t ) = cos 2 π n t T
Mathematically,
F ( t ) = b 0 + b 1 g 1 ( t ) + b 2 g 2 ( t ) + + a 1 f 1 ( t ) + a 2 f 2 ( t ) + ,
where n is a non-negative integer and all of a i , b i are real.
Problem
Is there a similar proof for the real form of Fourier's Theorem as for the complex form? I couldn't get one on the internet or in any book. Does this mean that it can be derived from the complex form? If yes then how? Any help would be appreciated.

2022-04-18 Answered

(8,y) and (0,-6); slope: 0

The forms of linear equations are related to secondary Algebra, yet learning the simplest equations will help you as well to come up with answers to linear Algebra. Start with an analysis of forms of equations based on examples that have been provided. If it does not make much sense to you yet, start with the study of point-slope, standard, and the famous slope-intercept form analysis. These different forms of linear equations will start making things clearer if you look at the answers that have been provided. For example, slope-intercept forms of equations would be y=mx + b equation formula.