Basis perpendicular to a subspace spanned by 2 vectors

**mybrohshi5** · April 17th 2010, 01:29 PM

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

**dwsmith** · April 17th 2010, 01:48 PM

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?

**mybrohshi5** · April 17th 2010, 05:08 PM

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

**dwsmith** · April 17th 2010, 05:13 PM

If you can tell me what the answer is, I can tell you what one it is looking for.

**mybrohshi5** · April 17th 2010, 05:42 PM

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

$\begin{bmatrix}.\\.\\.\\.\end{bmatrix}, \begin{bmatrix}.\\.\\.\\.\end{bmatrix}$

**dwsmith** · April 17th 2010, 05:47 PM

Both would be of that form.

Use the Gram-Schmidt process.

Do you know it?

**mybrohshi5** · April 17th 2010, 05:57 PM

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?

**dwsmith** · April 17th 2010, 05:59 PM

You can normalize it if you would like.

**mybrohshi5** · April 18th 2010, 08:56 AM

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?

**dwsmith** · April 18th 2010, 09:03 AM

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?

**mybrohshi5** · April 18th 2010, 09:49 AM

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....

**mybrohshi5** · April 18th 2010, 09:55 AM

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

**zzzoak** · April 18th 2010, 02:40 PM

I got these
(-68, -16, 0, 1)
( -1, 0, 1, 0).

**mybrohshi5** · April 18th 2010, 04:24 PM

Originally Posted by zzzoak

I got these
(-68, -16, 0, 1)
( -1, 0, 1, 0).

What did you do to find those?

**dwsmith** · April 18th 2010, 06:33 PM

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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