Question

# Let U,V be subspaces of Rn. Suppose that U⊥V. Prove that {u,v} is linearly independent for any nonzero vectors u∈U,v∈V.

Vectors and spaces

Let U,V be subspaces of Rn. Suppose that $$U\bot V.$$ Prove that $$\{u,v\}$$ is linearly independent for any nonzero vectors $$u\in U,\ v\in V.$$

2021-03-03

Given that $$U\bot V$$ that is $$(u, v) = 0$$ for any $$u\in U$$ and $$v\in V$$. Let
$$au+bv=0$$
$$\Rightarrow(au+bv,u)=0 \Rightarrow a(u,v)+b(u,v)=0 \Rightarrow a(u,u)=0$$ (since u is a non zero vector) $$\Rightarrow a=0$$
Again
$$au+bv=0 \Rightarrow (au+bv,v)=0 \Rightarrow a(u,v)+b(u,v)=0 \Rightarrow b(v,v)=0$$ (since v is a non zero vector) $$\Rightarrow b=0$$
Therefore $$\displaystyle{a}{u}+{b}{v}={0}\to{a}={b}={0}$$. Hence $$\{u,v\}$$ is linearly independent.