True or False Determine whether the statement is true or fal

True or False Determine whether the statement is true or false, and circle the correct answer. Each question is worth 2 points. If none of the vectors u, v, w in Ropf^3 is a multiple of one of the other vectors, then u, v, w are linearly independent. T F If a system of linear equations has two different solutions, it must have infinitely many solutions. T F If an augmented matrix [A b] can be transformed by elementary row operations into reduced row operations into reduced row echelon form, then the matrix equation [A b] is consistent. Every matrix is row equivalent to a unique matrix in row echelon form. If matrices A and B are row equivalent, they have the same reduced row echelon form. If an augmented matrix [A B] is transformed into [C d] by elementary row operations, then the matrix equations Ax = b and Cx = d have exactly the same solution set. If an n times n matrix A has n pivotpositions, then the reduced row echelon form of A is the n times n identity matrix. In some cases, it is possible for two vectors to span Ropf^3. If u, v, w are linearly independent, then u, v, w are not in Ropf^2. -u is in Span{u, v}.

Solution

1. F. Since, for example let u = (3, -2, 4), v = (2, 5, 2) and w = (5, 3, 6), i.e. none of the vectors are multiples of other vectors but w = u + v, hence the vectors are linearly dependent.

2. T. If a system of linear equations have two different solutions (i.e. solution is not unique), then atleast one variable has arbitrary assignment of values, hence it has infinite solutions.

3. F. The condition for consisteny of is Rank [A b] = Rank [A], and not REEF ([A]) or REEF ([A b]).

4. F. In contrast to reduced row echelon form (RREF), in echelon form elements above pivot element can be non- zero. For example for the following example second and third matrice are both row echelon forms of initial matrix M, which in turn implies echelon form of a matrix is not unique. (Remark: RREF is unique for every matrix)

5. F. With the reasons given in Problem No. 4.

6. T. Elementary Row transformations do not alter the solution of system of equations.

7. T. A matrix can be reduced to I_n if it has n pivot positions.

8. F. R³ can not be spanned by less than three independent vectors.

9. F. If u, v, w are in R², then at least one of the three vectors of u, v, and w is linearly dependent on the other two vectors.

10. T. (-1) . u + 0 v = - u