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Does the set (1 + x, 3 - x^2, 1 + 4x + x^2) span P_2? If not, then write one vector as a linear combin other two vectors, then add one of the standard basis vectors of P_2 to the set to form a basis for P_2. Consider the transformation T: R^3 rightarrow R, T(x, y, z) = (x +2y + 3z, y + z, x 3y + 4z, x + z) (a) Find the image of v (2, 3, -5). (b) Find the preimage of w = (0, -1, -1, 2). (c) basis for the range of T (d) Find a basis for the kernel of T.
Show transcribed image text Does the set (1 + x, 3 - x^2, 1 + 4x + x^2) span P_2? If not, then write one vector as a linear combin other two vectors, then add one of the standard basis vectors of P_2 to the set to form a basis for P_2. Consider the transformation T: R^3 rightarrow R, T(x, y, z) = (x +2y + 3z, y + z, x 3y + 4z, x + z) (a) Find the image of v (2, 3, -5). (b) Find the preimage of w = (0, -1, -1, 2). (c) basis for the range of T (d) Find a basis for the kernel of T.