A Linear system is a system that makes the result of 1. and 2. equal. 1. For any function x1(t) that goes into the system and is multiplied by A is added to a function x2(t) that goes into the system and is multiplied by B so that the added result is z(t)
(X1(t) --> system --> *A)
+ (X2(t) --> system --> *B)
-----------------------------
z(t)
2. For any function x1(t) that is multiplied by A and added to any function x2(t) that is multiplied by B, of which then the whole goes into the system.
[Ax1(t) + Bx2(t)]--> system --> w(t)
So if w(t) = z(t) then the system is linear.
Example of a Linear System
Let the system be y(t) = 2x(t)+1
Let x1(t) = 1 Let x2(t) = n Let A = 1 Let B = 5
1.
(1 --> system = (2(1)+1) = 3 --> 3*1 = 3)
+ (n --> system = (2(n)+1) = 2n+1 --> 5(2(n)+1) = 10n+5)
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= 10n+8
2.
[1*1 + 5n = 5n+1] --> system = 2(5n+1)+1 = 10n+3
Note: this is supposed to be a linear system...but for the life of me, I can't figure out why the answers aren't equal. Got any ideas?
Example of Non Linear System
Lets say that the system is y(t) = e^x(t)
let x1(t) = 3 let x2(t) = 4 Let A = 2 Let B = 3
1. 3 --> system = e^3 --> *A = 2e^3 4 --> system = e^4 --> *B = 3e^4 ------------------------------------ z(t) = 2e^3 + 3e^4
2. [Ax1(t) + Bx2(t)] --> system --> w(t)
[2*3 + 3*4 = 18] --> system = e^18
So e^18 is != 2e^3 + 3e^4 therefore this system is not linear.
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