Homework 7

(additional problems)

Problem 1   Let $W$ be a subspace of a finite-dimensional inner product space $V$. Show that $(W^{\perp })^{\perp }=W$, that is, the orthogonal complement of orthogonal complement of a subspace is the subspace itself.

Problem 2   Describe explicitly all inner products on $\mathbb{R}^{1}$ and on $\mathbb{C}^{1}$.

Problem 3   a) Let $T$ be the linear operator on $\mathbb{R}^{2}$ defined by $T((x_{1},x_{2}))=(-x_{2},x_{1})$ (that is, $T$ is rotation by $\pi /2$). Find all inner products $(\cdot ,\cdot )$ on $\mathbb{R}^{2}$ such that $(v,T(v))=0$ for all $v\in \mathbb{R}^{2}$. (Hint: start by noticing that the standard inner product $(\cdot ,\cdot )_{0}$ satisfies this condition).

b) Let $(\cdot ,\cdot )_{0}$ be the standard inner product on $\mathbb{C}^{2}$. Prove that there is no non-zero linear operator on $\mathbb{C}^{2}$ such that $(v,T(v))=0$ for all $v\in \mathbb{C}^{2}$.