Linear Algebra Review Problems for First Exam
Note: This practice test is longer than the actual exam–The actual exam will total 100 pts,
with roughly the same values as assigned here. Hopefully actually this isn’t too much review.
1.
(12 points) Find all solutions to the following system of linear equations
1 2

1
2 1
1
7 5
2
x
y
z
=

3
1
0
2.
(10 points) Let
V
be the set of all 2
×
2 matrices.
(a) Explan why
V
forms a vector space, describe a basis for V, and ﬁnd dim
V
.
(b) Let S be the set of all matrices
±
a b
c d
²
such that
a
+
b
+
c
= 0 and
a
+
d
= 0. Show that
S is a subspace of
V
, ﬁnd its dimension, and ﬁnd a basis for
S
.
3.
(12 points) (a) Let
v
1
, v
2
, ..., v
n
be a subset of a vector space
V
. A
linear combination
of
v
1
, v
2
, ..., v
n
is.
..
(b)Is (1
,
0
,
1
,
2) a linear combination of (2
,
1
,
3
,
0)
,
(7
,
3
,
1
,

1)
,
(0
,
1
,
4
,
3)?
4.