Question

Determine whether the set of all third-degree polynomial functions as given below, with the standard operations, is a vector space. If it is not, then determine the set of axioms that it fails. ax^3 + bx^2 + cx + d, a notequalto 0 Verity W = {(x, y, 3x-8y): x and y are real numbers} is a subspace of V = R^2. Determine whether the set S = {(-5, 6), (3, 7)} is linearly independent or linearly dependent. Determine whether the set S = {(3, 1), (-1, 3)} spans R^2. Explain why S = {(6, -5), (12, -10)} is not a basis for R^2. Find the rank of the matrix [-4 3 0 -24 18 0 24 -18 0 11 -5 1]. Find a basis for the subspace of R^3 spanned by S = {(16, 4, 30), (8, 2, 15), (4, 1, 5)}. Find the length of the vector v = (2, 4, 2). Find the distance d between u = (3, 3, 6) and v = (-3, 3, 0). Find the angle theta between the vectors u = (0, 5, 0, 5) and v = (3, 2, 7, 3). Find the vector v with length 3 and the same direction as the vector u = (-5, 5, 1).

Show transcribed image text Determine whether the set of all third-degree polynomial functions as given below, with the standard operations, is a vector space. If it is not, then determine the set of axioms that it fails. ax^3 + bx^2 + cx + d, a notequalto 0 Verity W = {(x, y, 3x-8y): x and y are real numbers} is a subspace of V = R^2. Determine whether the set S = {(-5, 6), (3, 7)} is linearly independent or linearly dependent. Determine whether the set S = {(3, 1), (-1, 3)} spans R^2. Explain why S = {(6, -5), (12, -10)} is not a basis for R^2. Find the rank of the matrix [-4 3 0 -24 18 0 24 -18 0 11 -5 1]. Find a basis for the subspace of R^3 spanned by S = {(16, 4, 30), (8, 2, 15), (4, 1, 5)}. Find the length of the vector v = (2, 4, 2). Find the distance d between u = (3, 3, 6) and v = (-3, 3, 0). Find the angle theta between the vectors u = (0, 5, 0, 5) and v = (3, 2, 7, 3). Find the vector v with length 3 and the same direction as the vector u = (-5, 5, 1).