Let \(L: R^{3} \rightarrow R^{3}\) be defined by $\mathrm{x}|\quad| \mathrm{x}+1 \mid$ \(\mathrm{L}|\mathrm{y}|=\mid 2 \mathrm{y}\) Show that \(L\) is not a linear transformation.

Let \(\mathrm{S}\) be the transformation defined by $\mathrm{S}(\mathrm{x}, \mathrm{y})=(2 \mathrm{x}+1\(, \)\mathrm{y}-2\( ). Show that \)\mathrm{S}$ is non- linear.

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 multiplication is associative, i.e., $(\mathrm{RS}) \mathrm{T}(\mathrm{v})=\mathrm{R}(\mathrm{ST}(\mathrm{v}))\(, where \)\mathrm{v} \varepsilon \mathrm{V}_{1}$

Let \(\mathrm{T}: \mathrm{R}^{4} \rightarrow \mathrm{R}^{4}\) be the linear transformation define by \(\mathrm{T}\left|\mathrm{x}_{1}\right|=\mid \quad 0\) $$ \left|\mathrm{x}_{2}\right| \quad\left|\mathrm{x}_{1}+\mathrm{x}_{2}\right| $$ \(\mathrm{x}_{3} \mid \quad \mathrm{x}_{4}\) \(\mathrm{x}_{4} \mid \quad 0\) Find the dimensions of \(\operatorname{Im}(\mathrm{T})\) and \(\operatorname{ker}(\mathrm{T})\).

Illustrate by means of an example that isomorphism, although an equivalence relation, is not a congruence relation.

Find the null spaces of the following linear transformations from \(\mathrm{V}\) to \(\mathrm{W}\). 1) The zero transformation 2) Any isomorphism \(\mathrm{T}: \mathrm{V} \rightarrow \mathrm{W}\) 3) The linear transformation associated with the following general system of linear equations: $$ \begin{aligned} &a_{11} x_{1}+a_{12} x_{2}=0 \\ &a_{21} x_{1}+a_{22} x_{2}=0 \end{aligned} $$