(e) T is the linear operator O Mzx2(R) defined byT(A) =3) (4-4)(E) Vis the vector space of polyuomilal functions in two real variables and y of degree at most 2 as ...

Let $T_{1}: V \rightarrow V$ be the dilation $T_{1}(v)=4 v .$ Find a linear operator $T_{2}: V \rightarrow V$ such that $T_{1} \circ T_{2}=I$ and $T_{2} \circ T_{1}=I$.

And this problem, we need to determine whether the given formula defines a 1-1 linear operator on em too. Now for this let us determine the kernel of the given linear operator, which is T. So that as we need to determine the kernel of teeth. Now, for that, let us consider any metrics E. In the kernel of people. Now let us pick a woman from this. T. A. Is equal to the zero matrix of order two by two because he belongs to the kernel of T. Now it is given that he is defined as E. So that will mean that he is equal to the zero matrix of order two by two. Now, since he's an elementary metrics, it will be in vertebral because every elementary matrix is in vertebral. So if we pretty multiply on both sides of the involves we have IAN was E. A. Is equal to E involves times the zero metrics. So in those times he will be the identity matrix, I and II times he will just be a any matrix times the zero matrix will be the zero matrix. So that means that E is equal to the zero matrix. So if any matrixes in qwerty, then this must be the zero matrix. So this would imply that the kernel of teak is the singleton set containing of only this zero matrix, and since the colonel only contains the zero vector of empty, too. Hence from here, we can conclude that linear operator P is 11