91Ó°ÊÓ

Determine the null space of the operator \(T: \mathbf{R}^{3} \longrightarrow \mathbf{R}^{2}\) represented by $$ \left[\begin{array}{rrr} 1 & 3 & 2 \\ -2 & 1 & 0 \end{array}\right] $$

Show that the set of all real numbers, with the usual addition and multiplication, constitutes a one-dimensional real vector space, and the set of all complex numbers constitutes a one-dimensional complex vector space.

Show that the norm \(\|x\|\) of \(x\) is the distance from \(x\) to 0 .

Show that \(\mathbf{R}^{n}\) and \(\mathbf{C}^{n}\) are not compact.

Let \(X\) and \(Y\) be normed spaces. Show that a linear operator \(T: X \longrightarrow Y\) is bounded if and only if \(T\) maps bounded sets in \(X\) into bounded sets in \(Y\).

Show that the operators \(T_{1}, \cdots, T_{4}\) from \(\mathbf{R}^{2}\) into \(\mathbf{R}^{2}\) defined by $$ \begin{aligned} &\left(\xi_{1}, \xi_{2}\right) \longmapsto\left(\xi_{1}, 0\right) \\ &\left(\xi_{1}, \xi_{2}\right) \longmapsto\left(0, \xi_{2}\right) \\ &\left(\xi_{1}, \xi_{2}\right) \longmapsto\left(\xi_{2}, \xi_{1}\right) \\ &\left(\xi_{1}, \xi_{2}\right) \longmapsto\left(\gamma \xi_{1}, \gamma \xi_{2}\right) \end{aligned} $$ respectively, are linear, and interpret these operators geometrically.

Show that the functionals defined on \(C[a, b]\) by $$ \begin{aligned} &f_{1}(x)=\int_{a}^{b} x(t) y_{0}(t) d t \\ &f_{2}(x)=\alpha x(a)+\beta x(b) \end{aligned} $$ are linear and bounded.

Give examples of compact and noncompact curves in the plane \(\mathbf{R}^{2}\).

If \(T \neq 0\) is a bounded linear operator, show that for any \(x \in \mathscr{I}(T)\) such that \(\|x\|<1\) we have the strict inequality \(\|T x\|<\|T\|\).

Find the dual basis of the basis \(\\{(1,0,0),(0,1,0),(0,0,1)\\}\) for \(\mathbf{R}^{3}\).