Let R^{4 }have the Euclidean inner product. Express w = (-1, 2, 6, 0) in the form w = w_{1 }+ w_{2}, where w_{1} is in the space W spanned by u_{1} = (-1, 0, 1, 2) and u_{2} = (0, 1, 0, 1), and w_{2 }is orthogonal to W
every vector in W are of the form
where and are in
since dimW1+dimW2=4
we know that dimW2 = 2. There are 2 vector x1 and x2 as <m,x1>=<m,x2> =0, and x1 and x2 are linearly independant.
let x1=(b,c,d,e)
so we can take x1=(1,0,1,0)
doing the same for x2 one that would work is x2=(0,1,2,-1)
there we've our 2 basic for w2 and w1 we could now try to find out to write the vector w given.
what we want is a basis for the orthogonal complement of W, W^{⊥} (we're already given the basis for W).
since every vector in W^{⊥} is orthogonal to all of W, this means:
(-1,0,1,2).(x,y,z,w) = 0
(0,1,0,1).(x,y,z,w) = 0
for any (x,y,z,w) in W^{⊥}.
from the second equation, we get: y+w = 0.
from the first, we get: -x+z+2w = 0.
since dim(W^{⊥}) = 2, we will have "two free parameters". we know that:
y = -w
x = z + 2w.
so if we select z and w, this will determine x and y.
so let's try z = 1, w = 0 for our first choice. this leads to (1,0,1,0). next, try z = 0, w = 1, this leads to (2,-1,0,1).
this gives us the basis {(-1,0,1,2),(0,1,0,1),(1,0,1,0),(2,-1,0,1)} for R^{4}.
since this IS a basis, we can write:
(-1,2,6,0) = a(-1,0,1,2) + b(0,1,0,1) + c(1,0,1,0) + d(2,-1,0,1), for some UNIQUE real numbers a,b,c,d.
that is:
(-1,2,6,0) = (-a+c+2d,b-d,a+c,2a+b+d)
or:
-a+c+2d = -1
b-d = 2
a+c = 6
2a+b+d = 0
we can solve this equation for a,b,c,d like we do any system of linear equations (form an augmented matrix, row-reduce, or any other method). since i'm lazy, and i hate row reduction with a passion, i will do this, instead:
b = d+2
a = 6-c
(c-6)+c+2d = -1
2(6-c)+d+2+d = 0
2c+2d = 5
-2c+2d = -14
4d = -9, so:
d = -9/4
c = 19/4
b = -1/4
a = 5/4
substituting back in, we have:
(-1,2,6,0) = (5/4)(-1,0,1,2) - (1/4)(0,1,0,1) + (19/4)(1,0,1,0) - (9/4)(2,-1,0,1), which you can verify.
thus w_{1} = (5/4)(-1,0,1,2) - (1/4)(0,1,0,1) = (-5/4,-1/4,5/4,9/4) <--this is in W,
and w_{2} = (19/4)(1,0,1,0) - (9/4)(2,-1,0,1) = (1/4,9/4,19/4,-9/4),
furthermore, w_{2} is orthogonal to W:
(-1,0,1,2).(1/4,9/4,19/4,-9/4) = -1/4 + 19/4 - 18/4 = 0
(0,1,0,1).(1/4,9/4,19/4,-9/4) = 9/4 - 9/4 = 0
since it is orthogonal to the basis vectors of W, and thus any linear combination of them (including w_{1}).