Let v1 2 1 1 v2 0 8 2 v3 6 5 1 Find the values of h for

Let v1 = [2 -1 1] , v2 = [0 8 -2], v3 = [ 6 5 1]. Find the values of h for which b = [h 2 3] is in the span {v1, v2, v3}.

Solution

v1 = [2 -1 1], v2 = [0 8 -2], v3 = [6 5 1] Find the value(s) of h for which b = [h 2 3] is in the span{v1, v2, v3}.

pv1 + qv2 + rv3 = b

p[2 , -1 , 1] + q[0 , 8 , -2] + r[6 , 5 , 1] = [h , 2 , 3]

[2p + 6r , -p + 8q + 5r , p - 2q + r] = [h , 2 , 3]

Equating :

2p + 6r = h
-p + 8q + 5r = 2
p - 2q + r = 3 --> multiply by 4 --> 4p - 8q + 4r = 12

-p + 8q + 5r = 2
4p - 8q + 4r = 12

Adding :

3p + 9r = 14
p + 3r = 14/3

First equation was :
2p + 6r = h
p + 3r = h/3

So, comparing these equations :

h/3 = 14/3

h = 14 --> ANSWER