Can anyone help me better understand Dr. Strang's "trick" in Lecture 22, just before 39:00, wherein here creates a matrix by "cheating?" One vector of the matrix is based on the equation F_k+1 + F_k = F_k+2,creating the vector [1,1] makes sense to me; but the "trick" vector F_k+1=F_k+1 leading to a vector of [1,0] doesn't make sense because it seems like the two vectors don't treat the unknowns or variables in the same way. Can someone try explaining it different to help me understand? Thanks in advance.

Question

Can anyone help me better understand Dr. Strang's "trick" in Lecture 22, just before 39:00, wherein here creates a matrix by "cheating?"  One vector of the matrix is based on the equation F_k+1 + F_k = F_k+2,creating the vector [1,1] makes sense to me; but the "trick" vector F_k+1=F_k+1 leading to a vector of [1,0] doesn't make sense because it seems like the two vectors don't treat the unknowns or variables in the same way.  Can someone try explaining it different to help me understand?  Thanks in advance.

phi · Answer

He introduces the column vector
$$u_{k}= \left[\begin{matrix}F_{k+1} \ F_k\end{matrix}\right]$$

and so (using the same pattern) it must be
$$u_{k+1}= \left[\begin{matrix}F_{k+2} \ F_{k+1}\end{matrix}\right]$$

We want some matrix A such that
$$ u_{k+1}= A u_k$$

we know the top element of $ u_{k+1}$ can be found with the top row of A being [1 1]
and the 2nd element of $ u_{k+1} $ is found by having the 2nd row of A being  [1 0 ]