I watched the (https://youtu.be/9HfA0o-_4u4) regarding this topic from the YouTube channel Physics videos by Eugene Khutoryansky. It helped me visualize how lateral numbers are useful to understand this phenomenon.
But it got me also confused about linear systems in general. Can somebody by starting of explaining what is a linear system in the first place and what are some real world engineering applications?
Thanks in advance.
In: Physics
*Linearity* in general essentially refers to the property that something (a function, system, relation, etc.) acts on the *whole* in a way determined solely by how it acts on the *parts*. For example, a function is linear when it obeys `f(a+b) = f(a) + f(b)`.
In the context of Fourier analysis, the central theme is that *any* function can be represented as a sum of *sine waves*. For example, let’s say we have some function `a(t) = A₁·sin(w₁·t) + A₂.sin(w₂·t) + …`. If some *linear* operator `F` acts on this function, that means the action of the operator on the function as a whole must be determined *solely* by how the operator acts on the individual sine waves of frequency ω₁, ω₂, and so on.
In other words, knowing how an operator acts on a sine wave of every frequency is sufficient to know how an operator acts on *any* function. This knowledge is called the *frequency response* of an operator, and it can be a useful tool to characterize or study the properties of linear operators acting on funbctions, which are otherwise somewhat hard to visualize intuitively.
(They can also be useful to simplify math in a lot of practical applications, for example signal theory – because it turns out that a convolution in the time domain is equal to a multiplication in the frequency domain, and multiplications are much easier to compute than convolutions for large signals)
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