# Subsets and subspaces, perpendicular sets!

• May 6th 2008, 10:49 PM
flawless
Subsets and subspaces, perpendicular sets!
M is a subset of R^n. M* is a set of vectors which are perpendicular to M (ie each element of M* is perpendicular to each element of M).

Firstly: Show that M* is a subspace of R^n

Secondly : In R^3, find M* where M = {2i-j+2k,i-2j-2k}
• May 6th 2008, 11:51 PM
Isomorphism
Quote:

Originally Posted by flawless
M is a subset of R^n. M* is a set of vectors which are perpendicular to M (ie each element of M* is perpendicular to each element of M).

Firstly: Show that M* is a subspace of R^n

Secondly : In R^3, find M* where M = {2i-j+2k,i-2j-2k}

Firstly:
Let $m \in M$,

(+) $\forall u,v \in M^{*}, m.(u+v) = m.u + m.v = 0 + 0 = 0 \Rightarrow u+v \in M^{*}$

(.) $\forall u \in M^{*}, \forall t \in \mathbb{R}, m.(tu) = t(m.u) = 0 \Rightarrow tu \in M^{*}$