Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Q. S is a subspace of R^3 containing only the zero vector. If S is spanned by (1,1,1) and (1,1,-1) what is a basis for S perp?

This is what I have so far -> a+b+c = 0 and a+b-c = 0. 2a+2b = 0

[-b] [-a] [0]

so a basis is [-1] [-1] [0]

I am confused. I am not sure what I am doing. Any help would be appreciated.

Thanks.

share|improve this question
    
Do you mean $S$ perp (as in perpendicular)? This space: $S^{\perp} = \{ x \in \Bbb{R}^3 \; \mid \; x \cdot y = 0 \text{ for all } y \in S \}$. –  Sammy Black Feb 27 at 7:59

2 Answers 2

up vote 0 down vote accepted

I think you mean $$S = \mbox{span}\{(1,1,1)\, (1,1,-1)\},$$ which is not the subspace containing only the zero vector. As far as finding the perp, what you have is almost there. A vector $(a,b,c) \in S^\perp$ must dot product with those two to zero giving the equations you obtained. And indeed, from those you get $2a + 2b = 0$. This means that $b = -a$, so we can choose $a = 1$, so that $b = -1$. Also, note that $c = 0$ because if you subtract the equations, you get $2c = 0 \rightarrow c = 0$. Therefore, $(1,-1,0)$ is a basis for $S^\perp$.

share|improve this answer
    
I can choose any number of a, right? (as long as it turns out to be linearly independent). Here's the exact question: If S is the subspace of R3 containing only the zero vector, what is S prep? If S is spanned by (1, 1, 1), what is S prep ? If S is spanned by (1, 1, 1) and (1, 1, -1), what is a basis for S prep ? –  Rami Feb 27 at 8:12
    
You can choose any number $a$ so that in general the vector is of the form $(a,-a,0)$. Note that this is three different questions: 1. $S = \{0\}$, 2. $S = \mbox{span}\{(1,1,1)\}$, and 3. $S = \mbox{span}\{(1,1,1),(1,1-1)\}$. –  Suugaku Feb 27 at 8:16
    
So, should the basis be (a,b,c) or (-b,-a,c), where c = 0. I am sort of confused. I am referring to my answer on line 4 of the original post. –  Rami Feb 27 at 8:25
    
The basis should be $\{(a,b,0)\}$, with $b = -a$, so $\{(a,-a,0)\}$ is a basis for the third question. –  Suugaku Feb 27 at 8:31

If $S$ contains only the zero vector, then $S$ is not spanned by $(1,1,1)$ and $(1,1,-1)$. Also, what does prep stand for?

share|improve this answer
    
S⊥ is the orthogonal subspace of S (pronounced S prep). Could you please explain a little why S cannot be spanned? –  Rami Feb 27 at 8:08
    
If $S$ is spanned by $(1,1,1)$ and $(1,1,-1)$, then it contains $(1,1,1)$. Since $S$ contains only the zero vector, this is a contradiction. –  5xum Feb 27 at 8:12

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.