91Ó°ÊÓ

Which of the following statements about linear vector spaces are true? Where a statement is false, give a counter-example to demonstrate this. (a) Non-singular \(N \times N\) matrices form a vector space of dimension \(N^{2}\). (b) Singular \(N \times N\) matrices form a vector space of dimension \(N^{2}\). (c) Complex numbers form a vector space of dimension \(2 .\) (d) Polynomial functions of \(x\) form an infinite-dimensional vector space. (e) Series \(\left\\{a_{0}, a_{1}, a_{2}, \ldots, a_{N}\right\\}\) for which \(\sum_{n=0}^{N}\left|a_{n}\right|^{2}=1\) form an \(N\)-dimensional vector space. (f) Absolutely convergent series form an infinite-dimensional vector space. (g) Convergent series with terms of alternating sign form an infinite- dimensional vector space.

Find the lengths of the semi-axes of the ellipse $$ 73 x^{2}+72 x y+52 y^{2}=100 $$ and determine its orientation.