Math 329
Elementary Linear Algebra
EXAM I
(1) Let Q= (4, −1, 2)and `be the line
passing through the origin in the di-
rection of d= (1, 2, 2).
(a) Let u=OQ. Find the projection
of uonto d.
(b) Compute the distance between Q
and `. (Hint: Consider u−projd(u).)
(2) Find an equation of the line `through
(−1, 5, 0)and (2, 1, 1)in the paramet-
ric form x=p+td;
(3) Let `be the line from (2).
(a) Find an equation of the plane pass-
ing through the origin and is per-
pendicular to `.
(b) Find an equation of the plane con-
taining the origin and `.
(4) Find the distance between
x=·1
1¸+s·−2
3¸
and
x=·5
4¸+t·2
−3¸.
(5) Find the reduced row echelon form of
the following matrix:
A=
1−1−1 2 1
2−2−1 3 3
−1 1 −1 0 −3
(6) Solve the following linear system:
w−x−y+2z =1
2w −2x −y+3z =3
−w+x−y= −3
(7) Let v1, . . . , v5denote the columns of
the matrix Afrom (5). Is
Span{v1, . . . , v5}=R3?
(8) Find the intersection of x+2y −z=3
and 2x +3y +z=1.
(9) Find the equation describing the fol-
lowing:
Span
1
2
3
,
−1
−1
0
(10) Determine whether the following sets
of vectors are linearly independent. If
linearly dependent, find a dependance
relation among the vectors.
(a)
3
1
0
,
−2
3
1
(b)
0
0
0
1
,
0
0
2
4
,
0
4
−9
4
,
1
0
−3
1
(c)
1
2
−1
,
−1
−2
1
,
−1
4
4
,
4
6
0
(11) Solve the following linear system.
x+2y −3z =9
2x −y+z=0
4x −y+z=4
(12) Use Gauss-Seidel method to obtain a so-
lution to the following system that is ac-
curate to within 0.001.
x−2y =3
3x +2y =1
1