I will help with the basic matrix.
From we get the matrix
lol, you miss those lectures too? I'm doing the same course.
The 'U to the power 'an upside down T' thing you speak of is the orthogonal complement of U. The orthogonal compliment of a subspace U of an inner product V is the set of all vectors in V that are orthogonal to every vector in U.
Know Q3 yet?
Another way to look at it is that if it were invertibable then for every quadratic function (quadratic function here means a polynomial of degree not more than 2):
would be a unique quadratic, but it is neither unique nor always a quadratic
(put to see that it is not always a quadratic)