# 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 one-to-one 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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