3u 2upsilon 2upsilon 4w 4u 3u is linearly dependent for a

{3u - 2upsilon, 2upsilon - 4w, 4u; - 3u} is linearly dependent for all choices of u, upsilon, and w. Every set of orthogonal vectors in R^n is linearly independent. If A = BC, then every solution to C_x = 0 is a solution to A_x = 0. If A = BC, then R(A) R(B). Let A be an m x n matrix and B an n x p matrix. If C = AB, then {C_x | x R^p} {A_w | w R^n}. The linear span of the vectors (4, 0, 0, 1), (0, 2, 0, -1) and (4, 3, 2, 1) is the 3-plane x_1 - 2x_2 + 3x_3 - 4x_4 = 0 in R^4. The linear span of the vectors (4, 0, 0, 1), (0, 3, 2, 0) and (4, 3, 2, 1) is the 3-plane x_1 - 2x_2 + 3x_3 - 4x_4 = 0 in R^4. Let v, upsilon, and w be vectors in R^n. Then span{u, upsilon, w} = span{u + 2upsilon, u + 3upsilon, u + nu + w}. The set {x_1, x_2, x_3, x_4) | x_1 + x_3 = x_2, x_4 = 0} is a subspace of R^4. The set {x_1, x_2, x_3, x_4) | x_1 + x_3 - x_2 + x_4 = 1} is a subspace of R^4. Every subspace of R^n has an orthogonal basis. Ever subspace of R^n has an orthonormal basis. span {(1 2 3 4), (5 6 7 8), (9 10 11 12)} = span {(1 1 1 1), (0 1 2 3)}. Every subset of a linearly independent set is linearly independent. Let X and Y be subsets of R^n. If X Y and X is linearly dependent, then Y is linearly dependent. The row reduced echelon form of a square matrix is the identity if and only if the matrix is invertible. The vectors (2, 2, -4, 3, 0) and (0, 0, 0, 0, 1) are a basis for the subspace x_1 - x_2 = 2x_2 + x_3 = 3x_1 - 2x_4 = 0 of R^5. The vectors (1, 1), (1, -2), (2, -3) are a basis for the subspace of R^4 that is the set of solutions to the equations x_1 - x_2 = 2x_2 + x_3 = 3x_1 - 2x_4 = 0. {x R^5 | x_1 - x_2 = x_3 + x_4 = x_1 + 2x_4 = 1} is a subspace of R^5. The set {x R^4 | x_1 = x_2} {x R^4 | x_3 = x_4} is a subspace of R^4. The set {x R^4 | x_1 = x_2} n {x R^4 | x_3 = x_4} is a subspace of R^4. The equation A_x = b has a solution if and only if b is linear combination of the columns of A. The dimension of the column space of a matrix is equal to the dimension of its row space. If U, V, and W are subspaces of R^n so is {u + 2upsilon + 3w | u U, upsilon V and w W}. {(1, 1, 1), (1, 0, 1), (3, 2, 3)} is a basis for R^3. If {u, upsilon, w} is an orthonormal set of vectors, then u + upsilon + w and u + 2upsilon - 3w are orthogonal. The length of the vector (1, 2, 2, 4) is 9. There is a 2 times 3 matrix A and a 3 times 2 matrix B such that AB is the identity matrix. If A is a 2 times 3 matrix and B is a 3 times 2 matrix, then BA can not be the identity matrix because N(A) notequalto {0}.

Solution

(38). Since 4w -3u = -(3u-2v)-(2v-4w), hence the given vectors are linearly dependent. The statement is

         True.

(39) The statement is True.

(40) The statement is True (Ax = BCx = B(Cx) = B.0 = 0 if Cx = 0).

(41) The statement is False.

(45) The statement is True.

(50) The statement is True.

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