please help to answer these 8 questions, thank you~

Question 1. In each of the following, determine if the given set is a subspace of R2 or not. For each case in which the set is a subspace, verify that it is subspace of R2 by showing that (81), (32), and (S3) hold. For each case in which the set is not a subspace of R2, state one of the properties of a subspace that does not hold and give a counterexample showing that the property fails. a) U1: :eRz y=av2 b) U2: :eR'Z 3x+8y=0 5'7 2 c)U3= y ER 901120 Question 2. Let v be a fixed (but unknown) vector in R" and let U be the collection of all vectors x in R" that are orthogonal to v, that is, U = {xER" | x . v =0). a) Verify that U is a subspace of Rn. b) Now, let v = 2 E R3. For this v, U becomes 3 U = XERS x - [ 2 = 0 Find a basis B for U.Question 3. Consider the subspace U = Span(u1, u2, u3, u4, u5) of R4, Where 2 6 14 3 1 2 4 1 u1 _1 ,u2= _1 ,u3= 1 ,u4= 0 ,andu5= 1 1 5 3 a) Find a basis 13 for U consisting of a subset of {u1, u2, u3, u4, u5}. b) What is the dimension of U? Question 4. Let B be the linearly independent set of vectors B = u= V = -2 in R3. 2 4 a Let x = 1 Verify that x is in Span (B) and find [x]B. -2 b) If [y B = -6 , find y. c) Extend B to a basis of R3, that is, find a vector w such that C = {u, v, w} is a basis for R3.Question 5. Determine Whether the following are linear transformations (operators) from R2 into R2. If the map is a linear transformation, provide a proof that it is linear transformation (verify that (LTl) and (LT2) hold). If the map is a not linear transformation, state one of the properties of a linear transformation that does not hold (either (LTl), (LT2), or the 0 test) and give a counterexample showing that the property fails. x _ cos($) a: 0 b T = ) 2 y [963;] a: _ 7y Question 6. a) Let v be a (fixed, but unknown) vector in R". Consider the map T : R" - R" defined by T (u) = u . V V Show that T is a linear transformation by verifying that (LT1) and (LT2) hold. b) Let Ti : R2 -> R2 be the linear operator which orthogonally projects each vector in R2 onto the line ( in R2 with general equation 2x - 5y = 0. i) Find the standard matrix [Ti] of Ti. ii) Is Ti invertible? Justify your answer.Question 7. a) Let T : R3 > R2 be the linear transformation given by: T x _ 4x+3yz 3: _ x2y52 Find the standard matrix [T] of T. b) Let R : R2 > R2 be the linear operator that rotates each vector counterclockwise about the origin through an angle of g radians (i.e. 90) . Find the standard matrix [R] of R. 0) Let S : R3 > R2 be the composition S = R o T. Find the standard matrix [S] of S. Then use 1 [S] to compute S'(v), where v = 3 . l Question 8. Let T : R2 > R3 be a linear transformation such that: o The null space of the standard matrix [T] is the line in R2 with equation x 2y 2 0. 8 0T :31 4 2 Find the standard matrix [T] of T