Find an orthonormal basis for the subspace and extend it to an orthonormal basis for R4

Question

Aufgabe:

Find an orthonormal basis for the subspace < V₁, V₂, V₃ > of R⁴ , where
V₁ = (1,1,1,1)^Τ , V₂ = (3,1,1,3)^Τ , V₃ = (2,-2,-4,0)^Τ  and extend it to an orthonormal basis for R⁴.

Ansatz/Problem:

Ich kann die Orthonormalbasis für den Unterraum finden, aber ich weiß nicht, wie man diese zu einer Orthonormalbasis für R⁴ ergänzt. Welchen anderen Vektor muss man wählen? Welche Eigenschaften soll diese Vektor haben?

Answer 1

In order to find an orthonormal basis for the linear subspace generated by $v_1,v_2,v_3$, you may use the Gram-Schmidt process. $$\mathcal{B}^*_{\langle v_1,v_2,v_3\rangle }= \{v_1^*,v_2^*,v_3^*\}=\left\{\left(\begin{array}{c}\frac{1}{2}\\\frac{1}{2}\\\frac{1}{2}\\\frac{1}{2}\end{array}\right), \left(\begin{array}{c}\frac{1}{2}\\- \frac{1}{2}\\- \frac{1}{2}\\\frac{1}{2}\end{array}\right), \left(\begin{array}{c}\frac{1}{2}\\\frac{1}{2}\\- \frac{1}{2}\\- \frac{1}{2}\end{array}\right)\right\}$$ If you wish to extend that basis to an orthonormal basis of $\mathbb{R}^4$, you could add any vector $v_4$ that suffices the following two conditions:

$(\text{i})$ $\langle v_4,v_i^*\rangle =0$ where $i\in \{1,2,3\}$.

$(\text{ii})$ $\langle v_4,v_4\rangle =1$.

Try $v_4=(-0.5,0.5,-0.5,0.5)^T$.

Comments

thanks for the answer.
but in (i) <v₄, v_i> you mean the vectors v₁, v₂, v₃ above or those in B*.

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Those in $\mathcal{B}^*_{\langle v_1,v_2,v_3\rangle }$. I updated the answer.