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.

The commutator \([\mathrm{X}, \mathrm{Y}]\) of two matrices is defined by the equation $$ [\mathrm{X}, \mathrm{Y}]=\mathrm{XY}-\mathrm{Y} \mathrm{X} $$ Two anticommuting matrices \(A\) and \(B\) satisfy $$ \mathrm{A}^{2}=\mathrm{I}, \quad \mathrm{B}^{2}=\mathrm{I}, \quad[\mathrm{A}, \mathrm{B}]=2 i \mathrm{C} $$ (a) Prove that \(C^{2}=I\) and that \([B, C]=2 i A\). (b) Evaluate \([[[\mathrm{A}, \mathrm{B}],[\mathrm{B}, \mathrm{C}]],[\mathrm{A}, \mathrm{B}]]\).

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

Show that the quadratic surface $$ 5 x^{2}+11 y^{2}+5 z^{2}-10 y z+2 x z-10 x y=4 $$ is an ellipsoid with semi-axes of lengths 2,1 and \(0.5\). Find the direction of its longest axis.