# Basis perpendicular to a subspace spanned by 2 vectors

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• Apr 17th 2010, 01:29 PM
mybrohshi5
Basis perpendicular to a subspace spanned by 2 vectors
Find a basis perpendicular to W.

Let $V = \begin{bmatrix}1\\-4\\1\\4\end{bmatrix}$ and $U = \begin{bmatrix}3\\-13\\3\\-4\end{bmatrix}$

and let W be the subspace spanned by V and U.

I am pretty confused on this topic right now :( I just dont seem to know what i need to do to solve problems like these.

Any help would be great :)

Thank you
• Apr 17th 2010, 01:48 PM
dwsmith
Quote:

Originally Posted by mybrohshi5
Find a basis perpendicular to W.

Let $V = \begin{bmatrix}1\\-4\\1\\4\end{bmatrix}$ and $U = \begin{bmatrix}3\\-13\\3\\-4\end{bmatrix}$

and let W be the subspace spanned by V and U.

I am pretty confused on this topic right now :( I just dont seem to know what i need to do to solve problems like these.

Any help would be great :)

Thank you

Do you want an orthogonal basis or orthonormal basis?
• Apr 17th 2010, 05:08 PM
mybrohshi5
The question doesnt specify. It just says....... let W the subspace of ${\mathbb R}^4$ spanned by v and u. Find a basis of $W^{\perp}$.

I should probably know how to find both though im sure :)
• Apr 17th 2010, 05:13 PM
dwsmith
If you can tell me what the answer is, I can tell you what one it is looking for.
• Apr 17th 2010, 05:42 PM
mybrohshi5
Im not sure what the answer is. Its for online homework.

There are two blank 4x1 matrixes where i can put my answer in. so it looks like this

$\begin{bmatrix}.\\.\\.\\.\end{bmatrix}, \begin{bmatrix}.\\.\\.\\.\end{bmatrix}$
• Apr 17th 2010, 05:47 PM
dwsmith
Both would be of that form.

Use the Gram-Schmidt process.

Do you know it?
• Apr 17th 2010, 05:57 PM
mybrohshi5
Quote:

Originally Posted by dwsmith
Both would be of that form.

Use the Gram-Schmidt process.

Do you know it?

Yes i do. Is that all i have to do? Seems easy enough.

and will the Gram schmidt process give me an orthogonal basis correct? then from there i could find a orthonormal basis if i needed right?
• Apr 17th 2010, 05:59 PM
dwsmith
You can normalize it if you would like.
• Apr 18th 2010, 08:56 AM
mybrohshi5
Using the gram schmidt process is not getting me the right answer :(

I even normalized the orthogonal basis i got and those are wrong as well.

Using the process i got

$\begin{bmatrix}1\\-4\\1\\4\end{bmatrix}$ and $\begin{bmatrix}\frac{30}{17}\\-\frac{137}{17}\\\frac{30}{17}\\-\frac{152}{17}\end{bmatrix}$

i plugged those in and they were wrong.

so i normalized those to get the orthononormal vectors

$\frac{1}{\sqrt(34)}*\begin{bmatrix}1\\-4\\1\\4\end{bmatrix}$ and $\frac{1}{\sqrt(2569/17)}*\begin{bmatrix}\frac{30}{17}\\-\frac{137}{17}\\\frac{30}{17}\\-\frac{152}{17}\end{bmatrix}$

Can you see where i am going wrong?
• Apr 18th 2010, 09:03 AM
dwsmith
I just checked my dot product and it wasn't zero but your dot product is. What section in your book does this question relate too?
• Apr 18th 2010, 09:49 AM
mybrohshi5
This is for online homework and i dont have the book cause i am fairly good at math and the professor said we didnt need it unless we needed an extra source of help (mine is this forum :) haha).

If i look at the professors power points i would think it relates to the section called Orthogonal Vectors in R^n, but i am not positive....
• Apr 18th 2010, 09:55 AM
mybrohshi5
I found this on yahoo answers and it looks similar to my question but i am a little unsure what is done to get the answer. maybe you will know what is going on

Find a basis of the subspace of R4 that consists of all vectors perpendicular to both:? - Yahoo! Answers
• Apr 18th 2010, 02:40 PM
zzzoak
I got these
(-68, -16, 0, 1)
( -1, 0, 1, 0).
• Apr 18th 2010, 04:24 PM
mybrohshi5
Quote:

Originally Posted by zzzoak
I got these
(-68, -16, 0, 1)
( -1, 0, 1, 0).

What did you do to find those?
• Apr 18th 2010, 06:33 PM
dwsmith
When I followed that example, I obtained $x_3\begin{bmatrix}
-1\\
0\\
1\\
0
\end{bmatrix}+x_4\begin{bmatrix}
-68\\
-16\\
0\\
1
\end{bmatrix}$
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