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.

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The objective is to state whether the following statements are true or not and justify the answers by providing the appropriate facts or theorems or providing the counter examples if they are not true ... View the full answer