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Math 341 self quiz
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 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×n matrix and the equation Ax = b is consistent
for every b in R^{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 has a unique solution for each b in
R^{m}.
m. If an n×n matrixAhas 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 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^{5}.
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 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 v1, and v3 is
not a linear combination of v1 and v2. Then {v_{1}, v_{2},
v_{3}} 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 R^{5} onto R^{6}.
z. If A is an m×n matrix with m pivot columns, then the linear
transformation x → Ax is a onetoone mapping.
2. Let a an b represent real numbers. Describe the possible
solution sets of the (linear) equation ax =b. [Hint: The number of
solutions depends upon a and b
3. The solutions (x, y, z )of a single linear equation
ax + by + cz = d
form a plane in R^{3} when a, b, and c are not all zero.
Construct sets of three linear equations whose graphs (a) intersect in a
single line, (b) intersect in a single point, and (c) have no
points in common . Typical graphs are illustrated in the figure.
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