A question about orthgonal/orthonormal basis

[transgalactic]

i added the question in the link

http://img232.imageshack.us/my.php?image=img8282ef1.jpg

my problem with this question starts with this W(and the T shape up side down) simbol

it represents a vector which is perpandicular to W

so why are they ask me to find the orthogonal(perpandicular) basis
to that perpandicular to W vector??(its already perpandicular to W)

so my answer should be the vectors of W
but in the answer they extract the vectors
from the formula and look for a vector which is perpandicular
to both vectors of Wif there were only W then i whould exract the vectors of the formula
and using gramm shmit
i would find the orghonormal basis(which includes in itself orthogonality)

but i was ask to find the orthogonal vectors of this W (upsidedown T)

i don't know what is the formula of its vectors??

 

[EnumaElish]

Can you find vector(s) such that any and all vector(s) orthogonal to W can be expressed as a linear combination of these basis vectors?

 

[HallsofIvy]

i added the question in the link

http://img232.imageshack.us/my.php?image=img8282ef1.jpg

my problem with this question starts with this W(and the T shape up side down) simbol

it represents a vector which is perpandicular to W

so why are they ask me to find the orthogonal(perpandicular) basis
to that perpandicular to W vector??(its already perpandicular to W)

You seem to be interpreting "orthogonal basis" for $W^{\perp}$ as meaning vectors perpendicular to $W^{\perp}$! That's not correct. An "orthogonal basis" for a vector space, V, consists of vectors in V that are perpendicular to on another. For example, if the overall vector space is R³ and W is the z-axis, then $W^{\perp}$ is the xy-plane. An "orthonormal" basis for that is {(1, 0, 0), (0, 1, 0)}.

so my answer should be the vectors of W
but in the answer they extract the vectors
from the formula and look for a vector which is perpandicular
to both vectors of W


if there were only W then i whould exract the vectors of the formula
and using gramm shmit
i would find the orghonormal basis(which includes in itself orthogonality)

but i was ask to find the orthogonal vectors of this W (upsidedown T)

i don't know what is the formula of its vectors??

We can't answer that without knowing precisely what W is.