# Orthogonal complement by example

I know what orthogonal complement is. Let W be subset of inner product space V whose orthogonal complement we are considering.

Now, orthogonal complement of W is equal to orthogonal complement of orthogonal complement of orthogonal complement, i.e. W(perpendicular)=W(3×perpendicular)

I know how to prove it using definition. But I want to know how this is possible .help me to understand .

Also, why W is subset of W(2×perpendicular)?Not able to understand this.

Thanks!

• Start with $W = \{(1,0,0),(0,1,0)\}$ in $\mathbb{R}^3$ and start computing orthogonal complements - one, two and three times. – Ethan Bolker Apr 28 '17 at 18:09
• Is it possible to calculate ? Since vector space is infinite. – Ka Sikh Apr 28 '17 at 18:18
• Yes it is. Where will you find all the vectors perpendicular to both the vectors in $W$ in my comment? Think geometrically, not with algebra or formulas. – Ethan Bolker Apr 28 '17 at 18:50
• (1,0, 0) and (0,1,0) are perpendicular to z direction. 2nd time, z axis is perpendicular to xy plane . 3rd time, xy plane is perpendicular to z axis. Am I thinking right? – Ka Sikh Apr 28 '17 at 19:01
• Yes you are thinking correctly. – Ethan Bolker Apr 28 '17 at 21:30

## 1 Answer

Let W={(1,0,0),(0,1,0)} be subset of inner product space.

Think geometrically the situation. W (perpendicular )is thus z- direction. W (2×perpendicular) is xy plane . As we can see W is subset of W (2×perpendicular) here. Again ,W (3×perpendicular ) is z- direction which is equal to W (perpendicular).