91Ó°ÊÓ

Show that the zero mapping and the identity transformation are linear transformations.

A linear transformation, \(\mathrm{T}: \mathrm{V} \rightarrow \mathrm{W}\), is a function defined on a vector space \(\mathrm{V}\) over a field \(\mathrm{K}\) that satisfies i) \(\mathrm{T}\left(\mathrm{v}_{1}+\mathrm{v}_{2}\right)=\mathrm{T}\left(\mathrm{v}_{1}\right)+\mathrm{T}\left(\mathrm{v}_{2}\right)\) ii) \(\mathrm{T}\left(\alpha \mathrm{v}_{1}\right)=\alpha \mathrm{T}\left(\mathrm{v}_{1}\right)\) for \(\mathrm{v}_{1}, \mathrm{v}_{2} \varepsilon \mathrm{V}\) and \(\alpha \varepsilon \mathrm{K}\). Give examples of some non-linear functions by showing that they fail to satisfy either (i) or (ii).

Give examples of the following types of linear operators: a) two commutative operators b) two non- commutative operators.

Give examples of nilpotent operators.

1 ) Let \(\mathrm{V}\) be a vector space over a field \(\mathrm{F}\) and let \(\alpha \varepsilon \mathrm{F}\) be any nonzero scalar. Show that multiplication by \(\alpha\), \(\mathrm{M}_{\alpha}: \mathrm{V} \rightarrow \mathrm{V}\) is a vector-space isomorphism. 2) Let \(\mathrm{V}\) and \(\mathrm{w}\) be any vector spaces over \(\mathrm{F}\) and \(\mathrm{V} \varepsilon \mathrm{W}\) their Cartesian product. Show that \(\mathrm{V} \cong \mathrm{V} \times\\{0\\}\) where \(0 \varepsilon \mathrm{W}\).