Paper 1, Section II, B

Vectors and Matrices | Part IA, 2013

(a) Let λ1,,λd\lambda_{1}, \ldots, \lambda_{d} be distinct eigenvalues of an n×nn \times n matrix AA, with corresponding eigenvectors v1,,vd\mathbf{v}_{1}, \ldots, \mathbf{v}_{d}. Prove that the set {v1,,vd}\left\{\mathbf{v}_{1}, \ldots, \mathbf{v}_{d}\right\} is linearly independent.

(b) Consider the quadric surface QQ in R3\mathbb{R}^{3} defined by

2x24xy+5y2z2+65y=0.2 x^{2}-4 x y+5 y^{2}-z^{2}+6 \sqrt{5} y=0 .

Find the position of the origin O~\tilde{O} and orthonormal coordinate basis vectors e~1,e~2\tilde{\mathbf{e}}_{1}, \tilde{\mathbf{e}}_{2} and e~3\tilde{\mathbf{e}}_{3}, for a coordinate system (x~,y~,z~)(\tilde{x}, \tilde{y}, \tilde{z}) in which QQ takes the form

αx~2+βy~2+γz~2=1.\alpha \tilde{x}^{2}+\beta \tilde{y}^{2}+\gamma \tilde{z}^{2}=1 .

Also determine the values of α,β\alpha, \beta and γ\gamma, and describe the surface geometrically.

Typos? Please submit corrections to this page on GitHub.