1. Finding basis problem

Given S= [a,b,c,d] belongs to R4, a not = 0. Find a basis that is a subset of s.

I am unsure what the a not = 0 means.

Would it just be a(1,0,0,0)+b(0,1,0,0)+c(0,0,1,0)+d(0,0,0,1) a not = 0.

If not, can someone give me an example of the basis.

2. Originally Posted by firebio
Given S= [a,b,c,d] belongs to R4, a not = 0. Find a basis that is a subset of s.

I am unsure what the a not = 0 means.

Would it just be a(1,0,0,0)+b(0,1,0,0)+c(0,0,1,0)+d(0,0,0,1) a not = 0.

If not, can someone give me an example of the basis.

it is clear that $\displaystyle v_{1}=[a,0,0,0],v_{2}=[a,1,0,0],v_{3}=[a,0,1,0],v_{4}=[a,0,0,1]$ are subset of $\displaystyle S$.

now, we need to prove that $\displaystyle v_{1},v_{2},v_{3},v_{4}$ are linearly independent.
let $\displaystyle \lambda_{1}v_{1}+\lambda_{2}v_{2}+\lambda_{3}v_{3} +\lambda_{4}v_{4}=0$,
then we get $\displaystyle \lambda_{2}=\lambda_{3}=\lambda_{4}=0$
and $\displaystyle a\lambda_{1} +a\lambda_{2}+a\lambda_{3}+a\lambda_{4}=0$
because $\displaystyle \lambda_{2}=\lambda_{3}=\lambda_{4}=0$, then $\displaystyle a\lambda_{1}=0$.
because $\displaystyle a \not= 0$, then $\displaystyle \lambda_{1}=0$
hence, $\displaystyle v_{1},v_{2},v_{3},v_{4}$ are linearly independent.

3. Originally Posted by dedust
it is clear that $\displaystyle v_{1}=[a,0,0,0],v_{2}=[a,1,0,0],v_{3}=[a,0,1,0],v_{4}=[a,0,0,1]$ are subset of $\displaystyle S$.

now, we need to prove that $\displaystyle v_{1},v_{2},v_{3},v_{4}$ are linearly independent.
let $\displaystyle \lambda_{1}v_{1}+\lambda_{2}v_{2}+\lambda_{3}v_{3} +\lambda_{4}v_{4}=0$,
then we get $\displaystyle \lambda_{2}=\lambda_{3}=\lambda_{4}=0$
and $\displaystyle a\lambda_{1} +a\lambda_{2}+a\lambda_{3}+a\lambda_{4}=0$
because $\displaystyle \lambda_{2}=\lambda_{3}=\lambda_{4}=0$, then $\displaystyle a\lambda_{1}=0$.
because $\displaystyle a \not= 0$, then $\displaystyle \lambda_{1}=0$
hence, $\displaystyle v_{1},v_{2},v_{3},v_{4}$ are linearly independent.
probably you should check your vectors

4. Originally Posted by firebio
Given S= [a,b,c,d] belongs to R4, a not = 0. Find a basis that is a subset of s.

I am unsure what the a not = 0 means.

Would it just be a(1,0,0,0)+b(0,1,0,0)+c(0,0,1,0)+d(0,0,0,1) a not = 0.

If not, can someone give me an example of the basis.

That simply means vectors of the form [a, b, c, d] where a is not 0. But that also means that [0, 0, 0, 0] is not in the set nor is the scalar product of 0 with any member of the set. This is NOT a subspace of $\displaystyle R^4$ and so does not have a basis!

5. Originally Posted by HallsofIvy
That simply means vectors of the form [a, b, c, d] where a is not 0. But that also means that [0, 0, 0, 0] is not in the set nor is the scalar product of 0 with any member of the set. This is NOT a subspace of $\displaystyle R^4$ and so does not have a basis!
i thought the problem said that $\displaystyle S$ is a subset of $\displaystyle R^4$ and we need to find the basis for vector space $\displaystyle S$. I also didn't check whether it is closed under scalar multiplication or not.

6. Originally Posted by HallsofIvy
That simply means vectors of the form [a, b, c, d] where a is not 0. But that also means that [0, 0, 0, 0] is not in the set nor is the scalar product of 0 with any member of the set. This is NOT a subspace of $\displaystyle R^4$ and so does not have a basis!
if $\displaystyle a$ is any real,what would be the basis ?

7. Originally Posted by Raoh
if $\displaystyle a$ is any real,what would be the basis ?
if $\displaystyle a$ is any real, then $\displaystyle S = R^4$ and any basis for $\displaystyle R^4$ is also basis for $\displaystyle S$

8. Originally Posted by dedust
if $\displaystyle a$ is any real, then $\displaystyle S = R^4$ and any basis for $\displaystyle R^4$ is also basis for $\displaystyle S$
but when you answered his question you assumed that $\displaystyle a$ is any real,right ?

9. Originally Posted by HallsofIvy
That simply means vectors of the form [a, b, c, d] where a is not 0. But that also means that [0, 0, 0, 0] is not in the set nor is the scalar product of 0 with any member of the set. This is NOT a subspace of $\displaystyle R^4$ and so does not have a basis!
That was what i thought at first too... but i didnt think it was suppose to be the kind of question with a strange twist to it..

And a is any real number not = to 0?

10. Someone corrected my interpretation of this question. I though it was asking for a basis for the "subspace" S but in fact it is asking for a basis for $\displaystyle R^4$ consisting of vectors in the set S.

Try this: the standard basis for $\displaystyle R^4$ is <1, 0, 0, 0>, <0, 1, 0, 0>, <0, 0, 1, 0>, and <0, 0, 0, 1> where the last three are not from set S. Okay, just change the first component: try <1, 0, 0, 0>, <1, 1, 0, 0>, <1, 0, 1, 0>, and <1, 0, 0, 1>. Is that a basis for $\displaystyle R^4$? There are four vectors in it; are they independent?

11. Originally Posted by HallsofIvy
Someone corrected my interpretation of this question. I though it was asking for a basis for the "subspace" S but in fact it is asking for a basis for $\displaystyle R^4$ consisting of vectors in the set S.

Try this: the standard basis for $\displaystyle R^4$ is <1, 0, 0, 0>, <0, 1, 0, 0>, <0, 0, 1, 0>, and <0, 0, 0, 1> where the last three are not from set S. Okay, just change the first component: try <1, 0, 0, 0>, <1, 1, 0, 0>, <1, 0, 1, 0>, and <1, 0, 0, 1>. Is that a basis for $\displaystyle R^4$? There are four vectors in it; are they independent?
i think the standard basis of $\displaystyle R^4$ is also a basis of $\displaystyle S$.
no ?

12. Originally Posted by Raoh
but when you answered his question you assumed that $\displaystyle a$ is any real,right ?
any real but not 0
Originally Posted by HallsofIvy
Someone corrected my interpretation of this question. I though it was asking for a basis for the "subspace" S but in fact it is asking for a basis for $\displaystyle R^4$ consisting of vectors in the set S.

Try this: the standard basis for $\displaystyle R^4$ is <1, 0, 0, 0>, <0, 1, 0, 0>, <0, 0, 1, 0>, and <0, 0, 0, 1> where the last three are not from set S. Okay, just change the first component: try <1, 0, 0, 0>, <1, 1, 0, 0>, <1, 0, 1, 0>, and <1, 0, 0, 1>. Is that a basis for $\displaystyle R^4$? There are four vectors in it; are they independent?
yes,they are linearly independent, as I proved before