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 11 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 x n matrix and the equation Ax = b is consistent for some b, then the columns of A span R ^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 x n matrix and the equation Ax = b is consistent for every b in R ^m , then A has m pivot columns.

l. If an in x n matrix A has a pivot position in every row, then the equation Ax = b has a unique solution for each b in R ^m

m. If an n x n matrix A has n pivot positions, then the reduced echelon form of A is the n x n identity matrix.

n. If 3 x 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 x 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 Bare row equivalent m x n matrices and if the columns of A span R ^m , then so do the columns of B.

q. If none of the vectors in the set S = {v ₁ , v ₂ , v ₃ } in R 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 R ² .

s. In some cases, it is possible for four vectors to span R ^m .

t. If u and v are in R ^m , then -u is in Span {u, v}

u. If u, v, and w are nonzero vectors in R ² , then w is a linear combination of u and v

v. If w is a linear combination of u and v in T: R ⁿ , then u is a linear combination of v and w.

w. Suppose that v ₁ , v ₂ and v ₃ are in R ⁵ , v ₂ is not a multiple of v ₁ , and v ₃ is not a linear combination of v ₁ and v ₂ . Then {v ₁ , v ₂ , v ₃ } is linearly independent.

x. A linear transformation is a function.

y. If A is a 6 x 5 matrix, the linear transformation X ↣ Ax cannot map R ⁵ onto R ⁶ .

z. If A is an m x n matrix with m pivot columns, then the linear transformation x ↣ Ax is a one-to-one mapping.