A linear system is a system that gives a predictable output based on superposition. What this means is if you put the sum of two signals into a system, you can expect the output to be a combination of the two outputs if the inputs were placed into the system by themselves.

So for example, lets say you put signal x into the system and the output is Ax. Then you put signal y into the system and the output is By. Then a linear system with signals x and y as input at the same time should have an output of Ax + By.

An example of a linear system would be a hot dog vendor. One day you put $3 into the system and and the output is one hot dog. Then the next day you put $6 into the system and get two hot dogs out. Then the third day, you bring your girlfriend and put $9 into the system (because you're paying for her lunch too) and get three hot dogs out.


Mathematically if the input is x[n] and output is y[n] (this is discrete time because no one would let you buy half a hotdog), then the relationship between the two are:

 x[n] = y[n]  (where x[n] represents 3 dollars input, y[n] represents 1 hotdog being output)


    x1[n]->    y1[n]
     $3  ->  1 hotdog

    x2[n]->    y2[n]
     $6  ->  2 hotdogs

    x3[n] = x1[n] + x2[n]
     $9   =   $3  +  $6

      y3[n]     =   x3[n]
      3 hotdogs =    $9  

      y3[n]   =   y1[n]   +   y2[n]
    3 hotdogs = 1 hotdog  +  2 hotdogs


An example of a non-linear system would be if the hot dog vendor was having a sale of buy 3, get the 4th free. Like the previous example you would get 1 hot dog for $3, 2 hot dogs for $6, but you would get 4 hot dogs for $9.

 x[n] = y[n]  (where x[n] represents 3 dollars input, y[n] represents 1 hotdog being output)


    x1[n]->    y1[n]
     $3  ->  1 hotdog

    x2[n]->    y2[n]
     $6  ->  2 hotdogs

    x3[n] = x1[n] + x2[n]
     $9   =   $3  +  $6

      y3[n]     =   x3[n]
      4 hotdogs =    $9  

      y3[n]   =   y1[n]   +   y2[n]
    4 hotdogs = 1 hotdog  +  2 hotdogs

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Abstract algebra continues the conceptual developments of linear algebra, on an even grander scale.

Dr. Paul Garrett