Consider linear transformations mapping vectors in R2 to vectors in R, clearly these are maps from a 2D vectors space to a 1D one. Since the space mapped to is one dimensional, we can call the mappings linear functions. Let e,e be a basis for R. Clearly a linear function on f: R R can be defined for all v E R by giving its values for the basis vectors f(e), f(e). E.g. if f(e) = 2, f(e) = 5, then the vector 6e7e is mapped: f(6e - 7e) = 6f (e)-7 f(e) = 12 - 35 = -23. Using this property, let us define the two linear functions h, h, by h (e) = 1, h(e) = 0; h (e) = 0,h (e) = 1. For any linear function g: R R and any vector v E R, g(v) = g(e)h(v) + g(e)h (v). (i) Prove that this last equation holds, (ii) Let the basis for R be e = (2), e = (3) If your vectors are not independent, then adapt them. A linear function will be represented by a 1 2 matrix. Give matrices for: h, h in the standard basis and a matrix for the linear function defined by 6 (e) = 5,6 (e) = 6, again in the standard basis. Prove that h, h form a basis for the space of all linear functions using the result proved in (i) or otherwise. (iii)