MT210 QUIZ 2
İLKER S. YÜCE
FEBRUARY 8, 2011
Surname, Name:
QUESTION 1. §1.3 VECTOR EQUATIONS
Determine if bis a linear combination of a1,a2and a3. If it is, find the
weights c1, c2, c3so that c1a1+c2a2+c3a3=b.
b=
2
−1
3
,a1=
1
−2
0
,a2=
0
1
1
,a3=
5
−6
4
ANSWER 1.
We need to determine if the vector equation x1a1+x2a2+x3a3=bhas a
solution. We consider the augmented matrix [a1a2a3b]and find its reduced
echelon form.
1 0 5 2
−2 1 −6−1
0 1 4 3
2R1+R2↔R2
//
1 0 5 2
0 1 4 3
0 1 4 3
1 0 5 2
0 1 4 3
0 1 4 3
(−1)R2+R3↔R3
//
1 0 5 2
0 1 4 3
0 0 0 0
We see that
General Soluti on =
c1= 2 −5c3,
c2= 3 −4c3,
c3is f ree .
Since the vector equation has a solution, bcan be written as a linear combi-
nation of a1,a2and a3. For example, if c3= 0, then c2= 3 and c1= 2.
1