Let V12103 V23152 and V31021 Which of the following vectors

Let V1=(2,1,0,3) , V2=(3,-1,5,2), and V3=(-1,0,2,1).

Which of the following vectors are in span {V1,V2,V3}? If a vector is in the indicated span, then write it as a linear combination of the vectors above. If a vector is not in the indicated

span, then show that it cannot be written as a linear combination of the vectors above.

(a) (0,0,0,0) (b) (1,1,1,1)

Solution

a) (0,0,0,0) = a v1 +b v2 + c v3

=a(2,1,0,3) + b(3,-1,5,2) + c(-1,0,2,1).

0 = 2a+3b-c

0 = a-b

0 = 5b+2c

0 = 3a+2b+c

solving we get a = b =c = 0

we put in 4th equation ,which is satisfied too.

hence (0,0,0,0,) = 0v1 +0v3 + 0v3

b) (1,1,1,1) = a v1 +b v2 + c v3

=a(2,1,0,3) + b(3,-1,5,2) + c(-1,0,2,1).

1 = 2a+3b-c

1 = a-b

1= 5b+2c

1= 3a+2b+c

solving from 1st 3 equation we get

a = 14/15 , b = -1/15 ,c = 2/3

3a+2b+c = 10/3 ,which is not 1 ,so they cannot be written in linear combination.