91Ó°ÊÓ

Let \(\left\\{(1,-1,3),(0,1,-1),(0,3,-2)\right.\) be a basis of \(\mathrm{R}^{3}\). Find the dual basis \(\left(\varphi_{1}, \varphi_{2}, \varphi_{3}\right)\).

Let \(\mathrm{V}\) be the vector space of all polynomial functions from \(\mathrm{R}\) into \(\mathrm{R}\) with degree less than or equal to \(2 .\) Find a basis for \(\mathrm{V}\) by using the following procedures i) Find three linear functionals on \(\mathrm{V}\) ii) Use these functionals as a basis for \(\mathrm{V}^{*}\), the dual of \(\mathrm{V}\). iii) Use the functionals to find a basis for \(\mathrm{V}\).Annthilators, Transposes \& Adjoint

Show that the determinant of a \(2 \times 2\) matrix is a 2 - linear function and belongs to a subspace of all functions from \(\mathrm{V}\left(=\mathrm{K}^{\mathrm{nxn}}\right)\) into \(\mathrm{K}\).

Let \(\mathrm{F}\) be a field and let \(\mathrm{D}\) be any alternating 3-linear function on \(3 \times 3\) matrices over the polynomial ring \(F[\mathrm{x}]\). Let \(\quad \begin{array}{rlr} & \mathrm{x} & 0 & -\mathrm{x}^{2} \mid \\\ \mathrm{A}= & \mid 0 & 1 & 0 \mid \\ & 1 & 0 & \mathrm{x}^{3} \mid\end{array}\) Show that \(D(A)=\left(x^{4}+x^{2}\right) D\left(\varepsilon_{1}, \varepsilon_{2}, \varepsilon_{3}\right)\)