True or false. Please justify your answer. a. In some cases, it is possible for

Question

True or false. Please justify your answer. a. In some cases, it is possible for six vectors to span R5. b. If a matrix A is m n and if the equation Ax = b has a solution for some b, then the columns of A span Rm. c. If a system of linear equations has two different solutions, then it has infinitely many solutions. d. Every matrix is row equivalent to a unique matrix in echelon form. e. If v1 and v2 span a plane in R3 and if v3 is not in that plane, then {v1, v2, v3} is a linearly independent set.

Explanation / Answer

(a). Only a maximum of 5 linearly independent vectors can span R5. If 6 vectors span R5, then the spanning set is linearly dependent. Since the statement does not mention anything about linear independence, it is True.

(b). The statement is true. I If the columns of an m × n matrix span Rm, then the equation Ax = b is consistent for each b in Rm ( In this case, b is a linear combination of the columns. Then the equation Ax = b is consistent).

(c) The statement is True. If Ax1 = b and Ax2 = b, then Ax1 – Ax2 = A(x1 – x2) = b b = 0. Then xp = x1 – x2 is a non-trivial solution of homogeneous equation Ax = 0. Hence infinitely many solutions can be generated by choosing the constant c arbitrarily in x = c(x1 – x2) + x1.

(d) The statement is False. The RREF of a matrix is not unique.

( e ) ) The statement is True. If v3 is not in the plane spanned by v1 and v2, it cannot be a linear combination of v1 and v2.Then {v1, v2, v3} is a linearly independent set.

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