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Suppose A = [-2 1 -1 1 0 3 9 2 1 2 8 4]. (a) Find the rank and nullity of A. (b) Find a basis for the column space of A that consists of columns of A. (c) Find a basis for the null space of A. (d) Find a basis for the row space of A (e) Find a basis for the row space of A that consists of rows of A. Suppose A = [-2 4 -6 1 -1 1 -2 3 1 5 3 -6 9 2 12]. (a) Find the rank and nullity of A. (b) Find a basis for the column space of A that consists of columns of A. (c) Find a basis for the null space of A. (d) Find a basis for the row space of A. (e) Find a basis for the row space of A that consists of rows of A. Does the set {1 + x. 3 = x^2, 1 + 4x + x^2) span P_2? If not, then write one vector as a linear combination of the other two vectors, then add one of the standard basis vectors of P_2 to the set to form a bask for P_2. Consider the transformation T: R^3 rightarrow R^4. T(x, y, z) = (x + 2y + 3x, 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).
Show transcribed image text Suppose A = [-2 1 -1 1 0 3 9 2 1 2 8 4]. (a) Find the rank and nullity of A. (b) Find a basis for the column space of A that consists of columns of A. (c) Find a basis for the null space of A. (d) Find a basis for the row space of A (e) Find a basis for the row space of A that consists of rows of A. Suppose A = [-2 4 -6 1 -1 1 -2 3 1 5 3 -6 9 2 12]. (a) Find the rank and nullity of A. (b) Find a basis for the column space of A that consists of columns of A. (c) Find a basis for the null space of A. (d) Find a basis for the row space of A. (e) Find a basis for the row space of A that consists of rows of A. Does the set {1 + x. 3 = x^2, 1 + 4x + x^2) span P_2? If not, then write one vector as a linear combination of the other two vectors, then add one of the standard basis vectors of P_2 to the set to form a bask for P_2. Consider the transformation T: R^3 rightarrow R^4. T(x, y, z) = (x + 2y + 3x, 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).