Let K = C. Classify the following affine varieties up to isomorphism V (T;T-), V(T{ +T3), V(T† – Tỷ – 1), V(T† +Tỷ – T3) A'\ {0}, V(T2 – Tỉ,T, – Tỷ) \ {0}, where the first four varieties are subsets of A2 and the last is a subset of A3. In other words, determine which of them are isomorphic and which are not.

2. Is there a linear map from R³ to R³ such that T(1,3,0) = (0,0,0) and T(-2,0,0) = (0,0,0) and T(3, 7, 1) = (3, 2, -3)?

5. Show that the function that sends an ordered pair ((x1, x2, x3), (Y1, Y2, Y3)) of elements in R³ to x1 y1 + x33 does not define an inner product on R³.

2. Let : RR and y: R+ → R be mappings with 6(x) -2e-x. (a) Find the range of y. = 1. Use your answer to determine whether y is surjective. (b) Use the definition to show that & is bijective. (c) Find & oy. (d) Find the values of x for which 82(x) = 1. - and y(x) = 4 [3] [3] [2] [2]

8. Decide which of the following sets are linearly independent in R. Justify your answer in each case. (a) X₁ = {(1,0), (0, 1)} CR² (b) X₂ = {(1,0), (2,0)} CR² (c) X3 = {(-1,0), (0,0)} CR² (d) X₁ = {(1,0), (0, 1), (1, 1)} CR² (e) X5 = {(1,0,0), (0, 1, 0), (1, 1, 1)} CR³ (f) X6 = {(1,0,0), (0, 1, 0), (1, 1, 1), (-1,0,1)} CR³ (g) X7 = {(0, 0, 0), (0, 1, 0), (1, 1, 1), (-1,0,1)} CR³ (h) Xs = {(0, 1, 0), (1,1,1)} CR³ (i) X9 = {(4,3,0,0), (0, 0, 1, 1), (0, 0, 0, 1), (1, 0, 0, 1), (0, 1, 0, 1)} CR¹ (j) X10 = {(1,2,0,0), (0, 2, 3,0), (0, 0, -1, 1)} CR¹

8. Decide which of the following sets are linearly independent in R". Justify your answer in each case. (a) X₁ = {(1,0), (0, 1)} ℃ R² (b) X₂ = {(1,0), (2,0)} ℃ R² (c) X3 = {(-1,0), (0, 0)} CR² 2 (d) X₁ = {(1,0), (0, 1), (1, 1)} ℃ R² C (e) X5 = {(1,0,0), (0, 1, 0), (1, 1, 1)} ℃ R³ (f) X6 = {(1,0,0), (0, 1, 0), (1, 1, 1), (−1, 0, 1)} ℃ R³ (g) X7 = {(0, 0, 0), (0, 1, 0), (1, 1, 1), (−1, 0, 1)} ℃ R³ (h) Xs = {(0, 1, 0), (1, 1, 1)} ℃ R³ 8 (i) X9 = {(4,3,0,0), (0, 0, 1, 1), (0, 0, 0, 1), (1, 0, 0, 1), (0, 1, 0, 1)} ℃ R4 (j) X10 = {(1, 2, 0, 0), (0, 2, 3, 0), (0, 0, −1, 1)} ℃ Rª

5. Show the span{(1,0, 1), (-1,2, 3), (0, 1, – 1)} is all of R'.

Show that the map T : R3 -→R? given by T (\, Х2, Хз) %— (х1 + X Xз) is an оpen map.

4. Let A := {1, 2, 3, 4} := B. (a) What is the cartesian product of A and B. (b) Find S := {(a, b)|f € Z}. (d) Is S a Partial Order? Justify your answer.

6. The covariance between X and Y is calculated and recorded as SXY (and sxy is a negative number). A linear transformation is performed on X. The linearly transformed variable is called W, where W; = 5 + Xi. The covariance between W and Y is calculated and recorded as sWY. Which statement is correct? A) В) In this example, swy would be less than sxY. In this example, swY Would be equal to sxY. In this example, swy would be greater than SXXY. 7. The covariance between X and Y is calculated and recorded as sxy (and sxy is a positive number). A linear transformation is performed on X. The linearly transformed variable is called W, where W; = (4)X;. The covariance between W and Y is calculated and recorded as sWY. Which statement is correct? In this example, sxY would be less than swy. In this example, sxY Would be equal to swy. In this example, sxY would be greater than sWY. A) B) 8. The correlation between X and Y is calculated and recorded as rxY (and rxy is positive). A linear…