Reference no: EM131064546
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 X n matrix and the equation Ax = b is consistent for some b, then the columns of A span Rm.
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 Rm, 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 Rm.
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 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 X n matrices and if the columns of A span Rm, then so do the columns of B.
q. If none of the vectors in the set S = {v1, v2, v3} in R3 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 R2.
s. In some cases, it is possible for four vectors to span R5.
t. If u and v are in Rm, then u is in Span{u, v}.
u. If u, v, and w are nonzero vectors in R2, then w is a linear combination of u and v.
v. If w is a linear combination of u and v in Rn, then u is a linear combination of v and w.
w. Suppose that v1, v2, and v3 are in R5, 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 X 5 matrix, the linear transformation x → Ax cannot map R5 onto R6.
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.