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== A linear function == | == A linear function == | ||
| − | we have seen is a function whose graph lies on a straight line, | + | |
| + | we have seen is a function whose graph lies on a straight line, | ||
and which can be described by giving its slope and its y intercept | and which can be described by giving its slope and its y intercept | ||
| + | |||
| + | == Linearity == | ||
| + | |||
| + | If both system yield the same output function, this is called a linear system. | ||
| + | |||
| + | |||
| + | == Prove == | ||
| + | |||
| + | |||
| + | |||
| + | y(t)=2x(t) | ||
| + | |||
| + | [1] | ||
| + | |||
| + | x1(t)--->[system]---->y1(t)=2x1(t)---->*a---(1) a*2*x1(t) | ||
| + | |||
| + | x2(t)--->[system]---->y2(t)=2x2(t)---->*b---(2) b*2*x2(t) | ||
| + | |||
| + | (1)+(2)= 2ax1(t)+2bx2(t) | ||
| + | |||
| + | |||
| + | |||
| + | [2] | ||
| + | |||
| + | x1(t)--->*a---(3) a*x1(t) | ||
| + | |||
| + | x2(t)--->*b---(4) b*x2(t) | ||
| + | |||
| + | (3)+(4)=a*x1(t)+b*x2(t) ---->[system]---->2(a*x1(t)+b*x2(t))=2ax1(t)+2bx1(t) | ||
| + | |||
| + | The results of [1] and [2] are the same. Thus, this is linear system. | ||
| + | |||
| + | |||
| + | |||
| + | y(t)=x(t)^2 | ||
| + | |||
| + | [3] | ||
| + | |||
| + | x1(t)--->[system]---->y1(t)=x1(t)^2---->*a---(5) a*x1(t)^2 | ||
| + | |||
| + | x2(t)--->[system]---->y2(t)=x2(t)^2---->*b---(6) b*x2(t)^2 | ||
| + | |||
| + | (5)+(6)= a*x1(t)^2+b*x2(t)^2 | ||
| + | |||
| + | |||
| + | |||
| + | [4] | ||
| + | |||
| + | x1(t)--->*a---(7) a*x1(t) | ||
| + | |||
| + | x2(t)--->*b---(8) b*x2(t) | ||
| + | |||
| + | (7)+(8)=a*x1(t)+b*x2(t) ---->[system]---->(a*x1(t)+b*x2(t))^2 | ||
| + | |||
| + | The results of [3] and [4] are not the same. Thus, this is non-linear system. | ||
Latest revision as of 05:03, 12 September 2008
A linear function
we have seen is a function whose graph lies on a straight line, and which can be described by giving its slope and its y intercept
Linearity
If both system yield the same output function, this is called a linear system.
Prove
y(t)=2x(t)
[1]
x1(t)--->[system]---->y1(t)=2x1(t)---->*a---(1) a*2*x1(t)
x2(t)--->[system]---->y2(t)=2x2(t)---->*b---(2) b*2*x2(t)
(1)+(2)= 2ax1(t)+2bx2(t)
[2]
x1(t)--->*a---(3) a*x1(t)
x2(t)--->*b---(4) b*x2(t)
(3)+(4)=a*x1(t)+b*x2(t) ---->[system]---->2(a*x1(t)+b*x2(t))=2ax1(t)+2bx1(t)
The results of [1] and [2] are the same. Thus, this is linear system.
y(t)=x(t)^2
[3]
x1(t)--->[system]---->y1(t)=x1(t)^2---->*a---(5) a*x1(t)^2
x2(t)--->[system]---->y2(t)=x2(t)^2---->*b---(6) b*x2(t)^2
(5)+(6)= a*x1(t)^2+b*x2(t)^2
[4]
x1(t)--->*a---(7) a*x1(t)
x2(t)--->*b---(8) b*x2(t)
(7)+(8)=a*x1(t)+b*x2(t) ---->[system]---->(a*x1(t)+b*x2(t))^2
The results of [3] and [4] are not the same. Thus, this is non-linear system.
