(I am counting as HW3 the homework exercises supervised by Sam Coskey during the two weeks following Isabel’s birth, and as HW4 the two written exercises assigned by Sam that I collected on March 17.)
This exercise is due April 7 at the beginning of lecture.
Provide a proof verifying that the function 
is a distance function.
A kind of library 
To clarify: Points in
are pairs 
where 
. For instance, to verify the triangle inequality one needs to prove that if 
are three points in 
, then 
. To do this, one needs to assign coordinates to 
, say 
, and proceed algebraically using the definition of 
given in the post.
(The high level reason why the function
defines a distance is that it comes from a norm on 
given by the quadratic form of a positive definite matrix, but rather than appealing to such machinery, I expect direct algebraic verifications.)