Question

if a1=(7,-1,4) a2=(4,0,1) and a3=(-3,5,0) and the set S={a1,a2,a3} is a basis for R3, how do you find the equation y=2a1-3a2+a3 (ps that is the answer, I just don't know how to get to it)

Answer 1

trying to find the linear combination y, they give the answer but they dont explain how they got to it

Answer 2

perhaps I am reading this wrong

Answer 3

do you know the coordinates of y?

Answer 4

y=(-1,3,5)

Answer 5

didnt think it was part of it, but they did mention that a little before

Answer 6

yes, that is the important bit. a1(x,y,z) = (7,-1,4) a2(x,y,z) = (4,0,1) and a3(x,y,z) = (-3,5,0)

Answer 7

right, so how do we find the linear combination?

Answer 8

so a linear combination of the x-coordinates of a1,a2,a3 should give the x- coordinate of y,
a linear combination of the y-coordinates of a1,a2,a3 should give the y- coordinate of y,
a linear combination of the z-coordinates of a1,a2,a3 should give the z- coordinate of y.

Answer 9

so 7m+4n-3p=-1
-m+5p=3
and4m+n = 5

Answer 10

ahh ok thanks a lot, then solve the matrix from there eh?

Answer 11

3 equations, 3 unknowns, solve to get m,n,p those will be the coefficients of the equation
y = ma1+na2+pa3

Answer 12

just one more thing....what did we do here? we had a coordinate, then we established the basis, and found another coordinate

Answer 13

did we change it from cartesian?

Answer 14

no its in Cartesian coordinates only.
all we did was to find a linear combination of the three points which gives the point y.

Answer 15

it follows that [y]s=(2,-3,1) in the text, this isn't a coordinate?

Answer 16

no, those are the coefficients of the linear combination.

Answer 17

i gotta run, hope that helped!

Answer 18

thanks!