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Unformatted text preview: value of number c vector w = belongs
1 2 0
2
5 4 −2
c
to the column space of A? (b) Find the most general solution of A = w for this value of c.
x
(c) Without further calculation, identify a basis for a row space of A (i.e., for C (AT )).
( pts) Solution: (a) Vector w belongs to C (A) precisely when system A = w has a
x
solution. Thus, we need to solve it. We will do it by reducing augmented matrix of this
system: 11
0
2
1
1
0
2
−2
r
1 0 −2
4
0 2
0
0 1
4 −4 × 1/2
1
2
−2 2
−2 → →
1 2
0
0 0
2
0 −1
r
1
2
−2 −2
r
0
0
5 4 −2
c
−51
r
0 −1 −2 c − 10
+2
r
00
0 c − 12
For this system to have the solution we need to have c − 12 = 0, that is, c = 12.
(b) We already have done all reduction work for this problem in part (a). The free variable
is x3 , since the third column is the only nonpivot column. From the ﬁrst reduced equation
x1 − 2x3 = 4, we get x1 = 4 + 2x3 . The second equation x2 + x3 =−2 2
gives x2 = − − 2x.
2
3 x1
4
−2...
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 Spring '08
 STAFF
 Math, Linear Algebra, Algebra

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