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
ANSWERED
asked 2021-03-02

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

Answers (1)

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.

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