1. ## Orthogonal sets

Is it possible to find a orthogonal set that is not linearly independent?

2. ## Re: Orthogonal sets

Originally Posted by icanc
Is it possible to find a orthogonal set that is not linearly independent?
An orthogonal subset $\displaystyle S$ of an euclidean space is not linearly independent if and only if $\displaystyle 0\in S$ .

3. ## Re: Orthogonal sets

More generally, if $\displaystyle v_1, v_2, v_3, ..., v_n$ are orthogonal vectors in an inner product space (we need the inner product to be able to define "orthogonal"), let $\displaystyle a_1v_+ a_2v_2+ a_3v_3+ \cdot\cdot\cdot+ a_nv_n= 0$ be a linear combination of the vectors that is equal to the 0 vector. Take the inner product of both sides of that with each vector in turn. For example, the inner product of each side with $v_1[/tex] would be \displaystyle v_1\cdot(a_1v_1+ a_2v_2+ a_3v_3+ \cdot\cdot\cdot+ a_nv_n)= a_1v_1\cdot v_1+ a_2v_1\cdot v_2+ a_3\v_1\cdot v_3+ \cdot\cdot\cdot+ a_nv_1\cdot v_n v_1\cdot 0. The right side is obviously 0 while on the left side, every product except [itex]v_1\cdot\v_1$ is 0. You did not say "orthonormal" so we cannot assume $\displaystyle v_1\cdot v_1= 1$ but it is some non-zero number (as FernandoRevill says, assuming the set does NOT contain the 0 vector), p, so we have $\displaystyle a_1p= 0$ and must conclude that $\displaystyle a_1= 0$.

4. ## Re: Orthogonal sets

I am still a little confused. So, let's say that $\displaystyle v_1=$ [3, 1, 1], $\displaystyle v_2=$ [-1, 2, 1], and $\displaystyle v_3$ = [-1/2, -2, 7/2]. They are all orthogonal, I checked. Could you give me a similar example that would make the set not linearly independent?

5. ## Re: Orthogonal sets

sure, just add the 0-vector [0,0,0].

the point being, if we have an orthogonal set in R^3 that is all non-zero vectors, say v1,v2,v3. then consider any linear combination:

u = av1 + bv2 + cv3. if this is 0, then <v1,u> = 0, since <v1,0> = 0 (here <x,y> is just the dot product in R^3).

but <v1,u> = <v1,av1> + <v1,bv2> + <v1,cv3> = a<v1,v1> + b<v1,v2> + c<v1,v3> = a<v1,v1> (by orthogonality).

but since <v1,v1> is not 0 (because v1 is not 0), we must have a = 0.

now show that <v2,u> = 0 implies b = 0, and that <v3,u> = 0 implies c = 0, showing {v1,v2,v3} is linearly independent.

the only assumption we make is that none of the v's are 0-vectors, so any orthogonal set of non-zero vectors in R^3 is linearly independent.

6. ## Re: Orthogonal sets

Originally Posted by Deveno
sure, just add the 0-vector [0,0,0].

the point being, if we have an orthogonal set in R^3 that is all non-zero vectors, say v1,v2,v3. then consider any linear combination:

u = av1 + bv2 + cv3. if this is 0, then <v1,u> = 0, since <v1,0> = 0 (here <x,y> is just the dot product in R^3).

but <v1,u> = <v1,av1> + <v1,bv2> + <v1,cv3> = a<v1,v1> + b<v1,v2> + c<v1,v3> = a<v1,v1> (by orthogonality).

but since <v1,v1> is not 0 (because v1 is not 0), we must have a = 0.

now show that <v2,u> = 0 implies b = 0, and that <v3,u> = 0 implies c = 0, showing {v1,v2,v3} is linearly independent.

the only assumption we make is that none of the v's are 0-vectors, so any orthogonal set of non-zero vectors in R^3 is linearly independent.
Thanks a lot, this helps a ton. One more question. Does adding the zero vector still make the set orthogonal?

7. ## Re: Orthogonal sets

yes, because the 0-vector is orthogonal to every other vector.

8. ## Re: Orthogonal sets

Originally Posted by Deveno
yes, because the 0-vector is orthogonal to every other vector.
Got it, thanks again.