Question 3.5: Show that P3 and M22 are isomorphic....

Consider the linear transformation T:P_{3}\to M_{22}, which is defined by

T\left(a x^{3}+b x^{2}+c x+d\right)={\left[\ \begin{array}{l l}{a}&{b}\\ {c}&{d}\end{array}\ \right]}.

Obviously, \dim(P_{3})=\dim(M_{22})=4.

Let T\left(a x^{3}+b x^{2}+c x+d\right)=\bar 0

\begin{array}{c}{\Rightarrow\left[{\begin{array}{c c}{a}&{b}\\ {c}&{d}\end{array}}\right]=\bar{0}}\\[1 em] {\Rightarrow a=0,\;b=0,\;c=0,\;d=0}\\[1 em] {\therefore{\mathrm{ker}}(T)=\{0\}.}\end{array}

\therefore T is one-one and onto.

\therefore P_{3} \cong M_{22}.

P_3 and M_{22} are isomorphic.