Description
Problem 1
(5+5 points)
a) Solve the following system of linear equations using the method taught in class.
−x1 + 2 x2 − x3 = 8
2 x1 + 3 x2 + 9 x3 = 5
−4 x1 − 5 x2 − 17 x3 = −7
b) Let v = (3, 1, 3)T be a vector expressed in coordinates with respect to the standard basis
of R
3
. Find the coordinates of this vector with respect to the basis
b1 =
1
1
0
, b2 =
1
0
1
, b3 =
0
1
1
.
Problem 2
(10 points)
Find conditions on α such that following system of linear equations has (a) exactly one
solution, (b) no solutions, or (c) an infinite number of solutions.
2 x1 − 2 x2 + α x3 = −2
4 x1 − 4 x2 + 12 x3 = −4
2 x1 + α x2 = 2
Problem 3
(5+5 points)
a) Determine whether the following vectors form a basis of R
4
. If not, obtain a basis by
adding and/or removing vectors from the set.
v1 =
2
0
0
1
, v2 =
−1
2
2
1
, v3 =
0
−1
2
−1
, v4 =
−2
2
1
0
.
b) A matrix is called singular if the homogeneous linear system Av = 0 has a “non-trivial”
solution v 6= 0.
Prove that AB = 0 implies that at least one of the matrices is singular.
