Can you express 14 17?23 as a linear combination of V?

V = (2 1 - 1), (1 - 1 2), (4 5 - 7)

No, we cannot express 14,17,23 as a linear combination of V

Step-by-step 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