Define the vector space analog of group homomorphism and ringhomomorphism. Such a mapping is called a linear transformation.Define the vector space analog of group isomorphism and ring isomorphism.

Which of the following mappings are linear transformations? Give a proof (directly using the definition of a linear transformation) or a counterexample in each case. [Recall that PÂ(F) is the vector space of all real polynomials p(x) of degree at most n with values in F.] • (-)-(~`.). = 3y z (ii) : P₂(F) → P4(F) given by o(p(x)) = p(x²) (so o(ax²+bx+c) = ax + bx² + c). (i) 0 : R³ → R² given by 0

Let T : R3 → R² bc given by a +b-2c -36 + c Tb

Let TR5 R³ be the linear transformation given by X1 x2 T x3 x4 X5 x1 + 2x2 x3 + 2x4x5 2x14x22x3 + 5x4 + x5 2x14x22x3 + 6x5 (a) Find the image of T and a basis for it. (b) Find the kernel of T and a basis for it. (c) Find the rank and nullity of T. Is the dimension of the domain of T equal to the sum of the rank and nullity that you found?