Answer:
No ,we cannot express 14,17,23 as a linear combination of V
Explanation:
Here 14,17,23 can be written as linear combination of V if we have a,b and c all non zero such that
(14,17,23) = a(2,1,-1)+b(1,-1,2)+c(4,5,-7)
(14,17,23) = (2a,a,-a)+(b,-b,2b)+(4c,5c,-7c) [ scalar multiplication ]
(14,17,23)= (2a+b+4c,a-b+5c,-a+2b-7c) [ vector addition ]
2a+b+4c = 14 equation 1
a-b+5c= 17 equation 2
-a+2b-7c =23 equation 3
Here we have three simultaneous equations
Since equation 2 and 3 has sign of a is opposite with coefficient 1,adding both equations,we get
b-2c = 40
multiplying equation equation 3 by 2 and adding it to equation 1 in order to eliminate the term a
2a+b+4c=14
-2a+4b-14c=46
adding 5b-10c = 60
divide it both sides by 5 ,we get
b -2c = 12
here we see that left side of two equations same but right side is not same
therefore solution doesnt exist .
Moreover given set of Vectors in V are not linear independent
since their determinant is zero
in order to write linear combination ,we need to have linearly independent vectors
there exist not values of a ,and c
therefore we cannot express 14,17,23 as linear combination of V