Chapter 2: Chapter 2

Let $\mathrm{T}: \mathrm{V}_{1} \rightarrow \mathrm{V}_{2} ; \mathrm{S}: \mathrm{V}_{2} \rightarrow \mathrm{V}_{3} ; \mathrm{R}: \mathrm{V}_{3} \rightarrow \mathrm{V}_{4}$ be linear transformations where \(\mathrm{V}_{1}, \mathrm{~V}_{2}, \mathrm{~V}_{3}\) and \(\mathrm{V}_{4}\) are vector spaces defined over a common field \(\mathrm{K}\). If we define multiplication of transformations by $$ \mathrm{S}^{\circ} \mathrm{T}(\mathrm{v})=\mathrm{S}(\mathrm{T}(\mathrm{v})) $$ show that the multiplication is associative, i.e., $$ (\mathrm{RS}) \mathrm{T}(\mathrm{v})=\mathrm{R}(\mathrm{ST}(\mathrm{v})), \text { where } \mathrm{v} \in \mathrm{V}_{1} $$

Show that in matrix arithmetic we can have the following: a) \(\mathrm{AB} \neq \mathrm{BA}\). b) \(\mathrm{A} \neq 0, \mathrm{~B} \neq 0\), and yet, \(\mathrm{AB}=0\). c) \(\mathrm{A} \neq 0\) and \(\mathrm{A}^{2}=0\) d) \(\mathrm{A} \neq 0, \mathrm{~A}^{2} \neq 0\), and \(\mathrm{A}^{3}=0\). e) \(\mathrm{A}^{2}=\mathrm{A}\) with \(\mathrm{A} \neq 0\) and $\mathrm{A} \neq \mathrm{I}$. f) \(A^{2}=I\) with \(A \neq-I\) and \(A \neq I\).

For \(\mathrm{n} \times \mathrm{n}\) matrices \(\mathrm{A}, \mathrm{B}\) show that det \(\mathrm{AB}=\operatorname{det} \mathrm{A}\) det \(\mathrm{B}\).

Find the volume of the parallelepiped determined by the vectors \(\mathrm{u}=(2,3,5), \mathrm{v}=(-4,2,6)\) and \(\mathrm{w}=(1,0,3)\) in \(\mathrm{xyz}\) -space.

Consider the vector space \(\mathrm{C}[0,1]\) of all continuous functions defined on \([0,1]\). If $$ \mathrm{f} \in \mathrm{C}[0,1] $$ show that $$ \left(1 \int_{0} \mathrm{f}^{2}(\mathrm{x}) \mathrm{dx}\right)^{1 / 2} $$ defines a norm on all elements of this vector space.

Let \(\mathrm{A}\) be an \(\mathrm{n} \times \mathrm{n}\) Hermitian matrix (i.e., $\mathrm{A}=\mathrm{A}^{*} \equiv\left(\mathrm{A}^{\mathrm{T}}\right)^{-}=\left(\mathrm{A}^{-}\right)^{\mathrm{T}}$ where \(\mathrm{A}^{-}\) is the conjugate of \(\mathrm{A}\) ). Show that the eigenvalues of \(\mathrm{A}\) are real.

Show that a linear operator \(\mathrm{A}\) on a finite-dimensional vector space \(\mathrm{X}\) is invertible if and only if it is one-to-one or onto.

Show that the \(\mathrm{n} \times \mathrm{n}\) matrix \(\mathrm{A}\) is invertible if and only if det \(\mathrm{A} \neq 0\)

Show that the system $$ \begin{aligned} &a_{11} x_{1}+\cdots+a_{1 n} x_{n}=b_{1} \\ &\cdot \\ &\cdot \\ &\cdot \\ &a_{n 1} x_{1}+\ldots+a_{n n} x_{n}=b_{n} \end{aligned} $$ has a unique solution if det \(\mathrm{A} \neq 0\) where Moreover, solve the system (1) for $\mathrm{x}_{1}, \ldots, \mathrm{x}_{\mathrm{n}}$ (Cramer's Rule).

Construct an orthogonal matrix from the eigenvectors associated with the symmetric matrix: $$ A=\begin{array}{rlr} 5 & 1 & 1 \\ 1 & 5 & -1 \mid \\ 1 & -1 & 5 \end{array} \mid $$ How does the transformation \(\mathrm{x}=\mathrm{Hy}\) affect the related quadric surface, where \(\mathrm{H}\) is the orthogonal matrix?