I'm trying to find multiple variables' invertible function I have two question,
If f(x, y) = (2x - y, x - 2y), (both real number)
what is invertible function of f(x, y)?
The answer is ( (2x-y)/3, ((x-2y)/3) what I don't know the way for this.
One more, if f(x, y, z) = (y + 2z, 3x - z, x + y)
the invertible function of this is ( (x+2y-z)/5, (-x-2y+6z)/5, (3x+y-3z)/5)
please help me for this...
So consider the function $$ f(x,y) = (2x-y, x- 2y) $$ It is a function of two inputs and two outputs. Let's write the outputs as $x_{\text{out}}$ and $y_{\text{out}}$, so we have $$ \tag{1} \left\{ \begin{array}{ccc} x_{\text{out}} &=& 2x&-y \\ y_{\text{out}} &=& x & -2y \end{array} \right. $$ In order to invert the process, we assume that we know the output values but not the input values. Because the function is linear, we can easily solve for the input values. Multiplying the second row in Equation $(1)$ by $-2$, we obtain $$ \left\{ \begin{array}{rcc} x_{\text{out}} &=& 2x&-y \\ -2y_{\text{out}} &=& -2x & +4y \end{array} \right. $$ and adding these together results in $$ x_{\text{out}} - 2y_{\text{out}} = 3y \qquad \Rightarrow \qquad y = \frac{x_{\text{out}}-2y_{\text{out}}}{3} $$ The input value $x$ can be obtained with a similar approach, by multiplying the first row by $-2$ in Equation $(1)$. The result is $$ \tag{1} \left\{ \begin{array}{rcc} -2x_{\text{out}} &=& -4x&+2y \\ y_{\text{out}} &=& x & -2y \end{array} \right. $$ Now the sum of the first two rows is $$ -2x_{\text{out}} + y_{\text{out}} = -3x \qquad \Rightarrow \qquad x = \frac{2x_{\text{out}}-y_{\text{out}}}{3} $$ Is it clear now?