5. With a pinch of linear algebra, the tangent space of a graph at a point can be written in a more computationally convenient form than Theorem 3.10.11. Here you will study this equivalent formulation. (5a) Begin by visualizing the special case of a graph in R3. Lemma A. Let the set S = {(x,y,z) R3 : 2 = F(x,y)} C R3 be the graph of a C1 function F: R2 R. Fix (a,b) R2. The tangent space of S at p=(a, b, F(a, b)) is given by = = = T S = span{(1,0,2,F(a,b)), (0, 1, O2F(a,b))}. Illustrate Lemma A with a detailed labelled picture of a transformation from R2 to R3. Use the function G:R2 R3 given by G(x,y)=(x,y, F(x,y)). Some quantities in your picture should be labelled using both G and F. You may label some objects with words, but do not write phrases or full sentences. Hint: Draw specific tangent vectors on the tangent plane. (5b) This lemma can be extended to higher dimensions. Lemma B. Let S C R be the graph of a Cfunction F : Rk Rn-k. Fix a Rk. The tangent space of S at p =(a, F(a)) is given by TPS = span{(ej, 0;F(a)):15j sk}. Do not prove this lemma. Let S CR5 be the graph of F(x,y,z) = (xe? + y, ye-+ x) and a = (2,1,0). Use Lemma B to express the tangent space T S at p = (a,F(a)) as a span of 3 vectors. 5. With a pinch of linear algebra, the tangent space of a graph at a point can be written in a more computationally convenient form than Theorem 3.10.11. Here you will study this equivalent formulation. (5a) Begin by visualizing the special case of a graph in R3. Lemma A. Let the set S = {(x,y,z) R3 : 2 = F(x,y)} C R3 be the graph of a C1 function F: R2 R. Fix (a,b) R2. The tangent space of S at p=(a, b, F(a, b)) is given by = = = T S = span{(1,0,2,F(a,b)), (0, 1, O2F(a,b))}. Illustrate Lemma A with a detailed labelled picture of a transformation from R2 to R3. Use the function G:R2 R3 given by G(x,y)=(x,y, F(x,y)). Some quantities in your picture should be labelled using both G and F. You may label some objects with words, but do not write phrases or full sentences. Hint: Draw specific tangent vectors on the tangent plane. (5b) This lemma can be extended to higher dimensions. Lemma B. Let S C R be the graph of a Cfunction F : Rk Rn-k. Fix a Rk. The tangent space of S at p =(a, F(a)) is given by TPS = span{(ej, 0;F(a)):15j sk}. Do not prove this lemma. Let S CR5 be the graph of F(x,y,z) = (xe? + y, ye-+ x) and a = (2,1,0). Use Lemma B to express the tangent space T S at p = (a,F(a)) as a span of 3 vectors