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Consider the algebraic transpose IT : Yf - Xy associated with the linear operator L : X - Y, where X and Y are linear vector spaces and Xy and Yf are their corresponding algebraic conjugates. Show that LT is linear.2. Let X and Y be vector spaces over the same set of scalars. Let LT[X, Y] denote the set of all linear transformations (i.e., functions) from X to Y. Thus, the transformations L and M in LT[X, Y] satisfy L(ar + =) = aL(x) + L(=) for all scalars a and vectors r, z E X. Define an addition operator between L and M as (L + M)(x) = L(x) + M(x), for all r E X. Define scalar multiplication by (aL)(x) = a(L(x)). Show that LT[X, Y] is a vector space.(10 points) Let B = {(1, -1), (3, 1) } and B' = {(1,2), (-1,0) ]. Find the transition matrix from B to B' 5. Let u = (1, -1, 0, 1, 1) and v = (0, 1, -2, 2, 1). (a) (5 points) Find u . v. (b) (5 points) Find d(u, v). 6. (10 points) Let be defined by = -Ujuju212. Is this an inner product? 7. (30 points) Let B = {(0, 3, 4), (1, 0, 0), (1, 1,0) }. Perform the Gram-Schmidt Process on B