The symmetric bilinear form \(\langle\cdot{,}\cdot\rangle_{\mathbf{A}}:\mathbb{R}^n\times\mathbb{R}^n\to\mathbb{R}\) is an inner product if and only if \(\det(\mathbf{A})\) and \(\operatorname{Tr}(\mathbf{A})\) are both positive.

• True
• False

The symmetric bilinear form \(\langle\cdot{,}\cdot\rangle_{\mathbf{A}}:\mathbb{R}^2\times\mathbb{R}^2\to\mathbb{R}\) is an inner product if and only if \(\det(\mathbf{A})\) and \(\operatorname{Tr}(\mathbf{A})\) are both positive.

• True
• False

In a Euclidean space, \(x+y\) and \(x-y\) are orthogonal if and only if \(\|x\|=\|y\|=1.\)

• True
• False

Let \(V\) be a Euclidean space and let \(x,y\in V.\) If \(\langle x,y\rangle = 0,\) then \(\|x\|^2+\|y\|^2=\|x+y\|^2.\)

• True
• False

Let \(V\) be a Euclidean space and let \(x,y,z\in V.\) If \(\langle x,y\rangle = \langle x,z\rangle,\) then \(y=z.\)

• True
• False

Let \(V\) be a Euclidean space and let \(\operatorname{span}\{x\}=U\subset V,\) where \(x\ne 0.\) If \(\langle x,y\rangle = \langle x,z\rangle,\) then \(y-z\in U^\perp.\)

• True
• False

Let \(V\) be a Euclidean space and let \(x,y\in V\) be orthogonal. Then \(x\) and \(y\) must be linearly independent.

• True
• False

Two distinct inner products on a vector space \(V\) can induce the same norm.

• True
• False

Consider a finite dimensional \(\mathbb{R}\)-vector space \(V\) equipped with an inner product \(\langle \cdot{,}\cdot\rangle.\) Given a linear map \(f\colon V \to \mathbb{R},\) there exists a \(v\in V\) such that \(f(u)=\langle u, v \rangle\) for all \(u\in V.\)

• True
• False

Given a subspace \(U\) of a finite dimensional Euclidean space \(V,\) we have that \(U=(U^{\perp})^{\perp}.\)

• True
• False