Answer to Question #143049 in Linear Algebra for Udy Eyo

Question

Express 5,9-4 as a linear combination of the vectors

V1= ( 2, 4, -5)

V2= (3, 7, -8)

V3= (7, 6, 1)

Answer 1

(5,9,-4) will be the linear combination of V1,V2 & V3 , then

(5,9,-4) = aV1 +bV2 + cV3

=> (5,9,-4) = a(2,4,-5) +b(3,7,-8) +c(7,6,1)

=> Now compare both side x,y,z component

then 2a+3b+7c = 5......................(1)

...also , 4a +7b +6c = 9......................(2)

...also -5a-8b+c = -4...................(3)

by (3)+(2)-(1) operation.... -3a-4b =0 => 3a= -4b ..........(4)

by (2)-6(3) operation.... 34a+55b = 33 .....(5)

Solving all above 4 and 5 equation , then putting these value of a and b in 1...

we have got $a= \frac{-132}{9} ,$ $b=\frac{99}{29} , c= \frac{16}{29}$ .

hence in terms of linear combination.....

(5,9,-4)= $\frac{-132V_1}{29} + \frac{99V_2}{29} + \frac{16V_3}{29}$