1.4 Answer Key
1. Let be a
matrix. Explain why the equation
cannot be consistent for all
in . Generalize your argument to the case of an arbitrary matrix with more rows than
columns.
Answer: can have at most four pivots. So, if we consider the augmented m

2.3 Answer Key
1. Let
for
be a
matrix such that
has a solution for every
to have a nontrivial solution? Justify your answer.
. Is it possible
Answer: Since is square and
has a solution for every
, by the Invertible
Matrix Theorem, is invertible. Using the

1.8 Answer Key
1. Define
by ( )
a. Show that is a linear transformation when
, ( )
Answer: When
(
Thus
)
(
.
. So, for any scalar
)
(
)
(
is a linear transformation when
and any
,
)
( )
.
b. Find a property of a linear transformation that is violated when

1.2 Answer Key
1. Suppose a 4x6 coefficient matrix has 4 pivot columns. Is the corresponding system of
equations consistent? Justify your answer.
Answer: Without loss of generality, the reduced echelon form of the matrix would be
[
]
Since we cant have a