Now we explore another (more abstract) method to define the tangent plane of a regular surface. In the following, S is always a regular surface and p S is a given point. Some linear algebra terminologies are listed at the end of this PSet. (8) (5pt) If S is the plane R?, show that F is a isomorphism. In other words, we can identify 7,5 and G. (When S is not R?, we need some new tools to prove this fact.) In summary, we can identify the tangent space and the space of good linear maps. Appendix. We recall some basic notions in linear algebra. Vector space. A vector space is a set X together two operators: . The addition, which is a map X x X -> X, sending element U E X and V E X to another element denoted by U + V. . The scalar multiplication, which is a map R x X - X, sending element a E R and V E X to another element denoted by aV. Moreover, these operators satisfy the following properties: . For any a, b E R and u, v, w E X, we have (a) (utv) tw= ut(vtw), (b) utv = vtu, (c) a(utv) = au + av, (d) (at b)u = au + bu, (e) (ab)u = a(bu), (f ) lu = u, (g) There exists an element 0 E X, called zero vector, such that for all u e X, Of u =u, (h) For any u E X, there exists another element, denoted by (-u), such that u+ (-u) =0. Now suppose we have two vector spaces X and Y. A map F : X - Y is called a homomorphism if it satisfies the following property: for any a, b E R and u, v E X, we have F(au + bv) = aF(u) + bF(v). A homomorphism F is called an isomorphism if F is bijective. There are also terminologies for injective and subjective homomorphisms