Compare the following solutions of problems from past assignments
with your private work. If you have not had a chance to do these
problems until now, please attempt to do them before looking
at these solutions.
Due September 12
No. 2
Put the first 3 columns of the augmented
matrix in reduced row echelon form:
(
1
5
-2
u
-2
4
-3
v
-1
-3
1
w
)
----->
(
1
0
{1}/{2}
- {3}/{2} u - {5}/{2} w
0
1
-{1}/{2}
{1}/{2} u + {1}/{2} w
0
0
0
v - 5 u - 7 w
)
(a) x = t (- {1}/{2}, {1}/{2}, 1) .
(b) There is no solution x.
(c) x = (-1, 0, 0) + t (- {1}/{2}, {1}/{2}, 1) .
(d) 5 y_{1} - y_{2} + 7 y_{3} = 0.
Due September 14
No. 1
Observe that
R(s, t) =
(
1
2
-2
-1
-2
2
)
(
s
t
)
,
and we put the first 2 columns of the augmented matrix in reduced row
echelon form:
(
1
2
x
-2
-1
y
-2
2
z
)
----->
(
1
0
- {1}/{3} x - {2}/{3} y
0
1
{2}/{3} x + {1}/{3} y
-2
2
z - 2 x - 2 y
)
(a) z = 2 x + 2 y .
(b) a plane.
Due September 24
No. 1
A few of the answers:
(a) x = t (2, -1, 1, 0) .
(b) x = (1, 0, 0, 0).
(d) x = (1, 1, 0, -2).
(g) The image of f is the set of all vectors y with
y_{1} = 2 y_{2} + 2 y_{3} + 9 y_{4} .