Exercise 5.3.3

Question

Complete an orthogonal system obtained in the previous problem to an orthogonal basis in $\mathbb{R}^3$, i.e., add to the system some vectors (how many?) to get an orthogonal basis.\par

Can you describe how to complete an orthogonal system to an orthogonal basis in general situation of $\mathbb{R}^n$ or $\mathbb{C}^n$?

Jing Huang · Accepted Answer

$
ewcommand{\T}{^\mathsf{T}}$
$
ewcommand{\mtbf}[2]{\mathbf{#1}_{#2}}$
	extbf{Solution} For 3D space, we already have 2 orthogonal vectors $\mathbf{v}_1, \mathbf{v}_2$ as the basis components. Then we just need another basis vector $\mathbf{v}_3$. The computation of $\mathbf{v}_3$ exploits the orthogonality, i.e., $(\mtbf{v}{1}, \mtbf{v}{3}) = 0, (\mtbf{v}{2}, \mtbf{v}{3}) = 0$. Expressed in matrix form, let $A = [\mtbf{v}{1}, \mtbf{v}{2}]$. Then solve $A\T \mtbf{v}{3} = \mathbf{0}$. (Since it is in 3D space, using cross product is also simple.)  \par
Generally, to complete an orthogonal system of $\mtbf{v}{1}, \mtbf{v}{2} ... \mtbf{v}{r}$. Consider $A = [\mtbf{v}{1}, \mtbf{v}{2}, ..., \mtbf{v}{r}]$, using the orthogonality, we compute the rest basis vectors by solving $A\T \mathbf{v = 0}$, i.e., the rest basis vectors compose an basis of Ker $A\T$ or Null $A\T$.

Exercise 5.3.3

Answers

Comments

Exercise 5.3.3

Answers

Comments

Add answer