Table of contents 
Introduction 
Fourier Transform 
Countable Infinity 
Dot Product (Inner Product) of Vectors 
Eigenvalue and Eigensignal 
Conclusion 
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A very basic concept in Signal and System analysis is the Transformation of signals. It involves a whole new paradigm of viewing signals in a context different from the natural domain of their occurrence.
Examples:
Example: Prove that the set of real numbers is not countably infinite.
 Suppose the set of real numbers is countably finite. Then every real number if mapped injectively onto the set of natural numbers.
 Let rk, where k N be the kth real number. Now we construct a real number r as follows: The integral part of r is 0.
 The kth decimal place of r is any integer that is different from the kth decimal place of rk.
 This number r which we have constructed differs from every rk at the kth decimal place. This contradicts our assumption that the set of real numbers is countably finite.
Note: A Discrete Signal x[n] can be thought of as a " Vector " with countably infinite dimensions. A Continuous Signal x(t) can be thought of as a vector with uncountably infinite dimensions.
Compare this with the definition of dot product for two finitedimensional vectors. We will now introduce two new terms  "Eigenvalue" and "Eigensignal". These concepts will be used later along with the concept of inner product of signals to introduce the Fourier series.
"Eigen" is a German word meaning "one's own".
In the context of Signals & Systems, eigen signals and eigenvalues are described as follows:
In this lecture you have learnt:
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