Question

Let U be the subspace of R^3 spanned by [1,-1,1]. Find a basis for the orthogonal complement of U.

Answer 1

The collection of all vectors in R^3 that are orthogonal to every vector in U is called the orthogonal complement of U.

Answer 2

you are essentially given a line, and asked to define the plane

Answer 3

Oh, okay.

Answer 4

more like a normal vector

Answer 5

So my line is {1,-1,1}

Answer 6

find the null space

Answer 7

but how can i find the nullspace if i don't have a matrix?

Answer 8

with the plane equation, you can then determine 2 vectors to form a basis from

Answer 9

or, just trial and error some dot products to fit

Answer 10

[1,-1,1] is a matrix

Answer 11

trial and error is prob the easiest way to do this one...you can do it in your head

Answer 12

oh okay so [1,-1,1][x1,x2,x3] = [0,0,0]

Answer 13

oh so wait do i have to find an orthogonal component first? and then find the basis of that?

Answer 14

does the basis have to be orthogonal as well, and unitized?

Answer 15

im just finding a basis for the orthogonal complement to U

Answer 16

x-y+z = 0

[1,1,0]
[0,1,1] would work for a trial and error

Answer 17

Then the orthogonal complement of W in V is the set of vectors u such that u is orthogonal to all vectors in W ( http://ltcconline.net/greenl/courses/203/Vectors/orthogonalComplements.htm)

seems like just 2 non-parallel vectors is all thats needed

Answer 18

oh okay so whichever one of those is a linear combination of [1,-1,1] is a basis for U orthogonal complement

Answer 19

i was getting the orthonormal basis mixed in my head :) an "orthonormal" basis is a basis where all the vectors in it are perpendicular as well. and have unit length.

Answer 20

oh okay yeah this is orthogonal haha