Linear interpolation formula In the following lecture: 

http://www.ece.mcmaster.ca/~xwu/interp_1.pdf

the model (formula) for solving the linear interpolation problem (1D) given at p.5 is:

$f(x)= a_1x_1 + a_0x_0$

solve for $a1,a0$

the system:

$f(0)=a_i*0 + a_0*1$

and

$f(1)=a_i*1 + a_0*1$

I don't understand where this formula comes from. Even though deriving in all possible ways I could think of, I cannot get the equation of a line $y=mx+b$  ( also written as: $f(x)= mx+b$, where m is the slope and b is the y-intercept) to look something like $f(x)= a_1x_1 + a_0x_0$. Any idea/proposals/discussions etc.?
To my eye, the equations in the slides look like $f(x) = a_1 x^1 + a_0 x^0 = a_1 x + a_0$. So it's an exponent on the $x$, not an index. As long as you are interpolating linear, the only exponents that you encouter are 1 (linear) and 0 (constant). When you go to quadratic, you'll see $x^2$ terms appearing. Now, $a_1$ is the coefficient of the linear term (your slope $m$) and $a_0$ is the coefficient of the constant term (the y-intercept $b$).

This should explain where the system of equations for $a_1$ and $a_0$ comes from.