thepillow
New member
- Joined
- Sep 12, 2012
- Messages
- 34
So I've been learning about orthonormal bases and I've come across a problem that's really stumping me (well, really just part 2 of the problem). I was hoping someone might be able to point me in the right direction.
Starting with this (non-standard) basis for R3 :
\[ B = \left\{\begin{bmatrix}1\\0\\1\end{bmatrix}, \begin{bmatrix}1\\1\\0\end{bmatrix}, \begin{bmatrix}1\\0\\-1\end{bmatrix} \right\} \]
part 1, which is pretty straight-forward, asks for an orthonormal basis N. Using Gram-Schmidt orthogonalization (and then dividing by the lengths to make them all unit vectors) I found
\[ N = \left\{\begin{bmatrix} 1\sqrt{2} \\0\\1/\sqrt{2}\end{bmatrix}, \begin{bmatrix}1/ \sqrt{6}\\ \sqrt{2/3}\\ -1/\sqrt{6}\end{bmatrix}, \begin{bmatrix}1/\sqrt{3} \\-1/\sqrt{3}\\-1/\sqrt{3}\end{bmatrix} \right\} \]
However, Part 2 is where I'm really stuck. The goal is to pick any two vectors from B and the corresponding two vectors from N and draw the four of them.
This usually wouldn't be too hard (even roughly representing 3D space on paper isn't too bad), but the twist here is that the problem asks for them to be drawn as if the piece of paper is the linear span of the 2 vectors chosen from B. For example, I could choose the first 2 vectors from B and N, and then I'd be drawing them as if the paper is the plane spanned by the set of 2 vectors:
\[\left\{\begin{bmatrix}1\\0\\1\end{bmatrix}, \begin{bmatrix}1\\1\\0\end{bmatrix}\right\}\]
I can't seem to wrap my head around this. It seems like it shouldn't be too hard to shift my perspective like this, but I've worked at it for a while and I'm really at a loss.
Thanks so much in advance for any advice, and sorry for the lengthy post.
Starting with this (non-standard) basis for R3 :
\[ B = \left\{\begin{bmatrix}1\\0\\1\end{bmatrix}, \begin{bmatrix}1\\1\\0\end{bmatrix}, \begin{bmatrix}1\\0\\-1\end{bmatrix} \right\} \]
part 1, which is pretty straight-forward, asks for an orthonormal basis N. Using Gram-Schmidt orthogonalization (and then dividing by the lengths to make them all unit vectors) I found
\[ N = \left\{\begin{bmatrix} 1\sqrt{2} \\0\\1/\sqrt{2}\end{bmatrix}, \begin{bmatrix}1/ \sqrt{6}\\ \sqrt{2/3}\\ -1/\sqrt{6}\end{bmatrix}, \begin{bmatrix}1/\sqrt{3} \\-1/\sqrt{3}\\-1/\sqrt{3}\end{bmatrix} \right\} \]
However, Part 2 is where I'm really stuck. The goal is to pick any two vectors from B and the corresponding two vectors from N and draw the four of them.
This usually wouldn't be too hard (even roughly representing 3D space on paper isn't too bad), but the twist here is that the problem asks for them to be drawn as if the piece of paper is the linear span of the 2 vectors chosen from B. For example, I could choose the first 2 vectors from B and N, and then I'd be drawing them as if the paper is the plane spanned by the set of 2 vectors:
\[\left\{\begin{bmatrix}1\\0\\1\end{bmatrix}, \begin{bmatrix}1\\1\\0\end{bmatrix}\right\}\]
I can't seem to wrap my head around this. It seems like it shouldn't be too hard to shift my perspective like this, but I've worked at it for a while and I'm really at a loss.
Thanks so much in advance for any advice, and sorry for the lengthy post.