Find a set of vectors that spans the subspace

In summary, to find a set of vectors in \mathbb{R}^3 that spans the subspace S\,=\,\{\,u\,\in\,\mathbb{R}^3\,|\,u\cdot v\,=\,0\,\}, where v=<1,2,3>, we can use the equation a+2b+3c=0 to find a set of vectors that are perpendicular to v and span this subspace. These vectors can be represented as <a, b, c>= <-2b- 3c, b, c>.
  • #1
mateomy
307
0
Find a set of vectors in [itex]\mathbb{R}^3[/itex] that spans the subspace
[tex]
S\,=\,\{\,u\,\in\,\mathbb{R}^3\,|\,u\cdot v\,=\,0\,\}
[/tex]
where v=<1,2,3>


Maybe 12 hours of studying is too much and I'm fried or, maybe I'm looking for excuses. Either way...

To solve this I'm trying to set up a matrix multiplication and augment it at zero. But, I just get a single linear equation which tells me that the only way I can have a span of this subspace is if my other vector is the zero vector <0,0,0>. I don't think that's right.

[tex]

\begin{bmatrix}
a & b & c
\end{bmatrix}

*

\begin{bmatrix}
1\\2\\3
\end{bmatrix}

=

\mathbf{0}


[/tex]

Getting [itex]a+2b+3c=0[/itex]

Where's my issue?


Thanks.
 
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  • #2
Perhaps you should stop and think what it is you're calculating here. You are looking for a subspace which is perpendicular to a vector, so it's a plane. Maybe that will help you interpret your result.
 
  • #3
From a+ 2b+ 3c= 0, you have a= -2b- 3c so <a, b, c>= <-2b- 3c, b, c>.
 
  • #4
HallsofIvy said:
From a+ 2b+ 3c= 0, you have a= -2b- 3c so <a, b, c>= <-2b- 3c, b, c>.

Thanks. I was looking at that and didn't take the next step of actually putting it into a vector form. Again, thanks.
 

1. What does it mean for a set of vectors to span a subspace?

A set of vectors spans a subspace if every vector in that subspace can be written as a linear combination of the vectors in the set.

2. How do I find a set of vectors that spans a given subspace?

To find a set of vectors that spans a subspace, you can use the basis and dimension theorem. This theorem states that any basis for a subspace will have the same number of vectors, known as the dimension, and any set of vectors with this same number of vectors will span the subspace.

3. Can a set of vectors that spans a subspace still contain redundant vectors?

Yes, a set of vectors that spans a subspace can contain redundant vectors. This means that some of the vectors in the set may be linearly dependent on others, and can be removed without changing the span of the subspace.

4. Is there only one set of vectors that can span a given subspace?

No, there can be infinitely many sets of vectors that span a given subspace. As long as the set of vectors has the same dimension as the subspace, it will span the subspace.

5. Can a set of vectors that spans a subspace also span a larger subspace?

Yes, a set of vectors that spans a subspace can also span a larger subspace. This is because the larger subspace will contain all the vectors in the smaller subspace, and the additional vectors in the set can still be written as linear combinations of the original set, making them span the larger subspace as well.

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