I was in a presentation recently given by a Facebook MySQL engineer and he mentioned there that using FORCE INDEX reduces I/O.

It was something to do with the MySQL Query Optimizer not needing to go to the data dictionary and make calculations. 


Can anyone explain how this works (preferably with references)?
Is this a good optimization practice to follow in general?



Question: Matrix Manipulation Question
Answer: To get more intuition for working with matrices, it might also help to consider a $n$ x $m$ matrix A as a representation of a linear transformation from an m-dimensional vector space $V$ to an n-dimensional vector space $W$, see: http://en.wikipedia.org/wiki/Linear_transformation 

A is not just a table of numbers with related operations; it produces elements of $W$ when fed with elements of $V$.

If you feed A with one $V$ element, it produces one $W$ element; if you feed it with two, it produces two. Also note that the linearity of the transformation implies (and is defined by) that for $v_1, v_2 \in V$ and $\lambda, \mu$ scalars, A fed with $\lambda v_1 + \mu v_2$ will produce $\lambda Av_1 + \mu Av_2$, thus: $A(\lambda v_1 + \mu v_2) = \lambda A(v_1) + \mu A(v_2)$