Solution: Mark each statement True or False. Justify each
Chapter , Problem 1E(choose chapter or problem)
Mark each statement True or False. Justify each answer. (If true, cite appropriate facts or theorems. If false, explain why or give a counterexample that shows why the statement is not true in every case.a. Every matrix is row equivalent to a unique matrix in echelon form.b. Any system of n linear equations in n variables has at most n solutions.c. If a system of linear equations has two different solutions, it must have infinitely many solutions.d. If a system of linear equations has no free variables, then it has a unique solution.e. If an augmented matrix [A b] is transformed into [C d] by elementary row operations, then the equations Ax = b and Cx = d have exactly the same solution sets.f. If a system Ax = b has more than one solution, then so does the system Ax = 0.g. If A is an m × n matrix and the equation Ax = b is consistent for some b, then the columns of A span ?m.h. If an augmented matrix [A b] can be transformed by elementary row operations into reduced echelon form, then the equation Ax = b is consistent.i. If matrices A and B are row equivalent, they have the same reduced echelon form.j. The equation Ax = 0 has the trivial solution if and only if there are no free variables.k. If A is an m × n matrix and the equation Ax = b is consistent for every b in ?m, then A has m pivot columns.l. If an m × n matrix A has a pivot position in every row, then the equation Ax = b has a unique solution for each b in ?m.m. If an n × n matrix A has n pivot positions, then the reduced echelon form of A is the n × n identity matrix.n. If 3 × 3 matrices A and B each have three pivot positions, then A can be transformed into B by elementary row operations.o. If A is an m × n matrix, if the equation Ax = b has at least two different solutions, and if the equation Ax = c is consistent, then the equation Ax = c has many solutions.p. If A and B are row equivalent m × n matrices and if the columns of A span ?m, then so do the columns of B.q. If none of the vectors in the set S = {v1, v2, v3} in ?3 is a multiple of one of the other vectors, then S is linearly independent.r. If {u, v, w} is linearly independent, then u, v, and w are not in ?2.s. In some cases, it is possible for four vectors to span ?m.t. If u and v are in ?m, then –u is in Span {u, v}u. If u, v, and w are nonzero vectors in ?2, then w is a linear combination of u and v.v. If w is a linear combination of u and v in T : ?n, then u is a linear combination of v and w.w. Suppose that v1, v2 and v3 are in ?5, v2 is not a multiple of v1, and v3 is not a linear combination of v1 and v2. Then {v1, v2, v3} is linearly independent.x. A linear transformation is a function.y. If A is a 6 × 5 matrix, the linear transformation x ? Ax cannot map ?5 onto ?6.z. If A is an m × n matrix with m pivot columns, then the linear transformation x ? Ax is a one-to-one mapping.
Unfortunately, we don't have that question answered yet. But you can get it answered in just 5 hours by Logging in or Becoming a subscriber.
Becoming a subscriber
Or look for another answer