Determine whether each of the following statements is True or False. If any item

Question

Determine whether each of the following statements is True or False. If any item is False, give a specific counterexample (write down a relevant set of vectors with numbers in them) to show that the statement is not always true. If v_1, v_2 and v_3, are in R^3 and v_3 is not a linear combination of v_1 and v_2, then {v_1, v_2, v_3} is linearly independent. If the vector equation c_1v_1 + c_2v_2 + c_3v_3 = 0, where the vectors v_k are in R^3, can only be solved with the constants c_1 = c_2 = c_3 = 0, then the set {v_1, v_2, v_3} is linearly independent. The set of vectors {v_1, v_2, v_3, v_4} R^3 is linearly dependent. If the set of vectors {v_1, v_2, v_3, v_4} R^4 is linearly dependent, then {v_1, v_2, v_3} is also linearly dependent. Let A = [- 2 1 3 6 1 - 13 -4 2 6]and b = [1 3 0], and consider the matrix transformation T(x) = Ax. For each of the following, answer "yes" or "no" and explain your reasoning: Is b in the domain of T? Is b in the codomain of T? Is b in the range of T? Consider the following transformation: T([x_1 x_2 x_3]) = [x_1 + 2x_3 4x_1 - 2x_2 + x_3 x_1 + x_3 2x_1 - x_3] What is the domain of T? What is the codomain of T? Show that T is a linear transformation by finding the standard matrix of the transformation.

Explanation / Answer

1- a) False. because if v3 = v1 + v2, then 1*v1 + 1*v2 -1*v3 = 0.

b) True

c) True

d) True

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