2
$\begingroup$

We have the real euclidean vector space $\mathbb{R}^3$ with the standard inner product and the standard basis $B = (e_1, e_2, e_3)$.

$W \subset \mathbb{R}^3$ is the subspace which is defined by: $$W = \{(x,y,z) \in \mathbb{R}^3 \mid 2x+y-z=0 \}$$

a.) Calculate an orthonormal basis of $W$.

I'm not sure how to solve this question. Do I have to use the Gram-Schmidt method?

How do I write out the matrix? Is this the matrix? $$ \begin{pmatrix} 2&1&-1 \end{pmatrix} $$

And then with the Gram-Schmidt algorithm: $$ w_1 = \frac{v_1}{\| v_1\|} = \frac{1}{\sqrt{5}}\begin{pmatrix} 2\\1\\-1\end{pmatrix} $$

Is my idea correct? If not, can you tell me where I went wrong?

Thank you!

$\endgroup$
  • $\begingroup$ No; you want to orthonormalise two linearly independent vectors satisfying $W$'s defining equation, so they have to be orthogonal to your vector. $\endgroup$ – J.G. Sep 7 '18 at 14:26
1
$\begingroup$

Hint

$W=\{(x,y,z)\in\mathbb{R}^3: 2x+y-z=0\}=\{(x,y,z)\in\mathbb{R}^3: z=2x+y\}=\{(x,y,z)\in\mathbb{R}^3: (x,y,z)=(x,y,2x+y)\}=\{(x,y,z)\in\mathbb{R}^3: (x,y,z)=(x,0,2x)+(0,y,y)\}=\{(x,y,z)\in\mathbb{R}^3: (x,y,z)=x(1,0,2)+y(0,1,1)\}=\langle(1,0,2),(0,1,1)\rangle$

$\endgroup$
1
$\begingroup$

First, you should find a basis of $W$ (Hint: $W$ has dimension $2$). Then you can apply Gram-Schmidt to this basis.

To find vectors of $W$, you want to look for solutions of $2x+y-z=0$. The vector $v=\begin{pmatrix} 2\\1\\-1 \end{pmatrix}$ you normalized is in fact the orthogonal complement of $W$: the condition $2x+y-z$ shows that $v$ is orthogonal to every vector of $W$.

If you know any vector $w\neq 0$ of $W$ (by guessing? It's not that hard), you can use the crossproduct $\times$ to find a third vector, which is already orthogonal to both $v$ and $w$, so you only have to normalize $w$ and $v\times w$ to obtain a orthonormal basis. (Note that this only works in $\mathbb R^3$)

$\endgroup$
  • $\begingroup$ This should be a comment. $\endgroup$ – blub Sep 7 '18 at 14:26
1
$\begingroup$

Start with any basis for $W$, such as $$\left\{\pmatrix{0 \\ 1 \\ 1}, \pmatrix{1 \\ 0 \\ 2}\right\}$$

and apply Gram-Schmidt to obtain the orthonormal basis $$\left\{\frac1{\sqrt2}\pmatrix{0 \\ 1 \\ 1}, \frac1{\sqrt3}\pmatrix{1 \\ -1 \\ 1}\right\}$$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.