is one member of the basis. Can you find the rest?
Problem:
Consider the vector
in
Find a basis of the subspace of R^4 consisting of all vectors perpendicular to .
I'm not very sure about how to solve this problem. I assume that I need to find a set of vectors for which the case holds.
I tried using the formula,
Since I already know , I would just need to figure out . To do this, I used projection.
This is where I get stuck, because I'm LOOKING for the u vectors... correct?
Am I far off the path I should be on here or am I on the right track?
Any help is greatly appreciated. Thanks for reading.
here is one way the vector (-2,1,0,0) might have been found:
we want the dot product (1,2,3,4).(x,y,z,w) to be 0. for our first vector, we can choose any x,y,z and w that will work. so we can choose z = w = 0,
and find x,y so that x + 2y = 0. if y = 1, x will have to be -2.
now, if you choose x = y = 0, clearly whatever vector you wind up with, will be linearly independent of (-2,1,0,0) (unless the only vector that works is (0,0,0,0)).
that will give you 2 elements of the basis, a similar approach will yield the 3rd.