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Find a set of vectors that spans the subspace

  • Thread starter mateomy
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  • #1
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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.
 

Answers and Replies

  • #2
938
9
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
HallsofIvy
Science Advisor
Homework Helper
41,772
911
From a+ 2b+ 3c= 0, you have a= -2b- 3c so <a, b, c>= <-2b- 3c, b, c>.
 
  • #4
307
0
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.
 

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