MATH 205A,B - LINEAR ALGEBRA
WINTER 2013
QUIZ 3
NAME: Section:(Circle one) A(1 : 10) B(2 : 40)
Show ALL your work CAREFULLY.
Let
A=
1−2−1
−1 2 2
2−4−1
.
(a) Express the solutions to the homogeneous equation A~x =~
0 in parametric form.
The coefficient matrix Acan be reduced as follows.
1−2−1
−1 2 2
2−4−1
∼
1−2−1
0 0 1
2−4−1
∼
1−2−1
0 0 1
0 0 1
∼
1−2−1
0 0 1
0 0 0
∼
1−2 0
001
000
.
The solutions to A~x =~
0are
~x =
x1
x2
x3
=
2x2
x2
0
=x2
2
1
0
where x2is the parameter.
(b) Based on your answer to (a), determine whether the columns of Aare linearly independent?
Justify your answer.
The columns of Aare NOT linearly independent since ~
0can be written as a non-trivial
linear combinations of the columns ~a1, ~a2,~a3. For instance, ~
0 = (2)~a1+ (1)~a2+ (0)~a3(take
x2= 1).
(c) Do the columns of Aspan R3? Explain.
The columns of Ado NOT span R3. For any vector ~
b=
b1
b2
b3
where b36= 0, the system
A~x =~
bdoes not have a solution because the row reduced echelon form of Ahas a row
of zeroes and thus the system would be inconsistent.
Date: January 25, 2013.
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