Linear Algebra - Gram-Schmidt Process

[joe_cool2]

Homework Statement

Let R² have the Euclidean inner product and use the Gram-Schmidt process to transform the basis {u₁,u₂} into an orthonormal basis.

u₁ = (1,-3)
u₂ = (2,2)

Homework Equations

Gram-Schmidt process:

<br /> \\v_1 = u_1<br /> \\v_2= u_2 -<br /> <br /> \frac{\left ( \left \langle u_2, v_1\right \rangle \right )}{\left \|<br /> v_1<br /> <br /> \right \| ^ 2}v_1<br />

The Attempt at a Solution

We'll go ahead and find the norm of v first. Then we'll evaluate the second Gram-Schmidt equation.

<br /> \\\left \| v_1 \right \| = \sqrt{10} \\\\<br /> \left \| v_1 \right \|v_1 = u_1 = (1,-3)<br /> \\\\\left \| v_2 \right \|v_2= \begin{bmatrix}<br /> 2\\2<br /> <br /> \end{bmatrix} -<br /> <br /> \frac{\left ( \left \langle \begin{bmatrix}<br /> 2\\2<br /> <br /> \end{bmatrix} , \begin{bmatrix}<br /> \frac{1}{\sqrt{10}}\\ \frac{-3}{\sqrt{10}}<br /> <br /> \end{bmatrix} \right \rangle \right )}{\left \| \begin{bmatrix}<br /> \frac{1}{\sqrt{10}}\\ \frac{-3}{\sqrt{10}}<br /> <br /> \end{bmatrix} \right \| ^ 2}<br /> <br /> \begin{bmatrix}<br /> \frac{1}{\sqrt{10}}\\ \frac{-3}{\sqrt{10}}<br /> <br /> \end{bmatrix}\\\\\left \| v_2 \right \|v_2= \begin{bmatrix}<br /> 2\\2<br /> <br /> \end{bmatrix} - \frac{\left ( \frac{-4}{\sqrt{10}} \right )}{1} \begin{bmatrix} \frac{1}{\sqrt{10}}\\ \frac{-3}{\sqrt{10}} \end{bmatrix}<br /> <br /> \\\\<br /> <br /> \begin{bmatrix}<br /> 2\\2<br /> <br /> \end{bmatrix}<br /> <br /> - \begin{bmatrix}<br /> \frac{-4}{10}\\\\\frac{12}{10}<br /> <br /> \end{bmatrix}<br /> <br /> =<br /> <br /> \begin{bmatrix}<br /> \frac{24}{10}\\\\\frac{8}{10}<br /> <br /> \end{bmatrix}<br /> <br /> =<br /> <br /> \begin{bmatrix}<br /> \frac{12}{5}\\\\\frac{4}{5}<br /> <br /> \end{bmatrix}<br />

I've not been able to get another answer for this step. Even checked in Matlab.

Now, we'll normalize the vector like the problem wants us to:
<br /> \\\left \| v_2 \right \| = \sqrt{8}<br /> \\<br /> v_2 = \begin{bmatrix}<br /> \frac{\frac{12}{5}}{\sqrt{8}}\\<br /> \frac{\frac{4}{5}}{\sqrt{8}}<br /> <br /> \end{bmatrix}<br /> <br /> =<br /> <br /> \begin{bmatrix}<br /> <br /> \frac{\frac{12}{5}}{\sqrt{8}}\\<br /> \frac{\frac{4}{5}}{\sqrt{8}}<br /> <br /> \end{bmatrix}<br /> <br /> =<br /> <br /> \begin{bmatrix}<br /> <br /> \frac{12}{10\sqrt{2}}\\\\<br /> \frac{4}{10\sqrt{2}}<br /> <br /> \end{bmatrix}<br /> <br /> =<br /> <br /> \begin{bmatrix}<br /> <br /> \frac{6}{5\sqrt{2}}\\\\<br /> \frac{2}{5\sqrt{2}}<br /> <br /> \end{bmatrix}<br /> <br /> =<br /> <br /> \begin{bmatrix}<br /> <br /> \frac{6\sqrt{2}}{10}\\\\<br /> \frac{2\sqrt{2}}{10}<br /> <br /> \end{bmatrix}<br /> <br /> =<br /> <br /> \begin{bmatrix}<br /> <br /> \frac{3\sqrt{2}}{5}\\\\<br /> \frac{\sqrt{2}}{5}<br /> <br /> \end{bmatrix}<br />

This is an odd problem, so I know that the solution should be v_2 = (\frac{3}{\sqrt{10}} , \frac{1}{\sqrt{10}}). There must be something wrong when I'm finding ||v_2||v_2 . Is there anything fundamentally wrong with my process?

 

[vela]

Homework Statement

Let R² have the Euclidean inner product and use the Gram-Schmidt process to transform the basis {u₁,u₂} into an orthonormal basis.

u₁ = (1,-3)
u₂ = (2,2)

Homework Equations

Gram-Schmidt process:

<br /> \\v_1 = u_1<br /> \\v_2= u_2 -<br /> <br /> \frac{\left ( \left \langle u_2, v_1\right \rangle \right )}{\left \|<br /> v_1<br /> <br /> \right \| ^ 2}v_1<br />

The Attempt at a Solution

We'll go ahead and find the norm of v first. Then we'll evaluate the second Gram-Schmidt equation.

<br /> \\\left \| v_1 \right \| = \sqrt{10} \\\\<br /> \left \| v_1 \right \|v_1 = u_1 = (1,-3)<br /> \\\\\left \| v_2 \right \|v_2= \begin{bmatrix}<br /> 2\\2<br /> <br /> \end{bmatrix} -<br /> <br /> \frac{\left ( \left \langle \begin{bmatrix}<br /> 2\\2<br /> <br /> \end{bmatrix} , \begin{bmatrix}<br /> \frac{1}{\sqrt{10}}\\ \frac{-3}{\sqrt{10}}<br /> <br /> \end{bmatrix} \right \rangle \right )}{\left \| \begin{bmatrix}<br /> \frac{1}{\sqrt{10}}\\ \frac{-3}{\sqrt{10}}<br /> <br /> \end{bmatrix} \right \| ^ 2}<br /> <br /> \begin{bmatrix}<br /> \frac{1}{\sqrt{10}}\\ \frac{-3}{\sqrt{10}}<br /> <br /> \end{bmatrix}\\\\\left \| v_2 \right \|v_2= \begin{bmatrix}<br /> 2\\2<br /> <br /> \end{bmatrix} - \frac{\left ( \frac{-4}{\sqrt{10}} \right )}{1} \begin{bmatrix} \frac{1}{\sqrt{10}}\\ \frac{-3}{\sqrt{10}} \end{bmatrix}<br /> <br /> \\\\<br /> <br /> \begin{bmatrix}<br /> 2\\2<br /> <br /> \end{bmatrix}<br /> <br /> - \begin{bmatrix}<br /> \frac{-4}{10}\\\\\frac{12}{10}<br /> <br /> \end{bmatrix}<br /> <br /> =<br /> <br /> \begin{bmatrix}<br /> \frac{24}{10}\\\\\frac{8}{10}<br /> <br /> \end{bmatrix}<br /> <br /> =<br /> <br /> \begin{bmatrix}<br /> \frac{12}{5}\\\\\frac{4}{5}<br /> <br /> \end{bmatrix}<br />

I've not been able to get another answer for this step. Even checked in Matlab.

If you get rid of a common factor of 4/5, you have $\vec{v}_2 = \begin{bmatrix} 3 \\ 1 \end{bmatrix}$.

Now, we'll normalize the vector like the problem wants us to:
<br /> \\\left \| v_2 \right \| = \sqrt{8}<br /> \\<br /> v_2 = \begin{bmatrix}<br /> \frac{\frac{12}{5}}{\sqrt{8}}\\<br /> \frac{\frac{4}{5}}{\sqrt{8}}<br /> <br /> \end{bmatrix}<br /> <br /> =<br /> <br /> \begin{bmatrix}<br /> <br /> \frac{\frac{12}{5}}{\sqrt{8}}\\<br /> \frac{\frac{4}{5}}{\sqrt{8}}<br /> <br /> \end{bmatrix}<br /> <br /> =<br /> <br /> \begin{bmatrix}<br /> <br /> \frac{12}{10\sqrt{2}}\\\\<br /> \frac{4}{10\sqrt{2}}<br /> <br /> \end{bmatrix}<br /> <br /> =<br /> <br /> \begin{bmatrix}<br /> <br /> \frac{6}{5\sqrt{2}}\\\\<br /> \frac{2}{5\sqrt{2}}<br /> <br /> \end{bmatrix}<br /> <br /> =<br /> <br /> \begin{bmatrix}<br /> <br /> \frac{6\sqrt{2}}{10}\\\\<br /> \frac{2\sqrt{2}}{10}<br /> <br /> \end{bmatrix}<br /> <br /> =<br /> <br /> \begin{bmatrix}<br /> <br /> \frac{3\sqrt{2}}{5}\\\\<br /> \frac{\sqrt{2}}{5}<br /> <br /> \end{bmatrix}<br />

This is an odd problem, so I know that the solution should be v_2 = (\frac{3}{\sqrt{10}} , \frac{1}{\sqrt{10}}). There must be something wrong when I'm finding ||v_2||v_2 . Is there anything fundamentally wrong with my process?

 

[joe_cool2]

Ok. So I gather that my answer is not technically wrong? How can I verify that my original answer is in fact an orthonormal basis spanning the same subspace?

I guess if I make computational errors, it will be unlikely that all the dot products in some generated set of vectors is zero. Right?

 

[vela]

Your answer is wrong. You didn't normalize the vector correctly.

 

[joe_cool2]

Ah, I see that I accidentally used the norm of u₂ instead of v₂ to normalize v₂. The book confused me by switching up u's and v's.

Checking to see if the norm of the final vectors is one would also be a useful trick.