MathAlgebraLinear algebraVectors and vector spaces

Euclidean spaces

Theory

A basis and an inner product

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Let VV be a two-dimensional vector space, {e⃗1,e⃗2}\left\{\vec{e}_{1},\vec{e}_{2}\right\} its basis. We know that for some arbitrary inner product ⟨e⃗1,e⃗1⟩=a1⟨e⃗2,e⃗2⟩=a2⟨e⃗1,e⃗2⟩=c\left\langle\vec{e}_{1},\vec{e}_{1}\right\rangle = a_{1}\\ \left\langle\vec{e}_{2},\vec{e}_{2}\right\rangle = a_{2}\\ \left\langle\vec{e}_{1},\vec{e}_{2}\right\rangle = cWhich of the following equalities are always true?

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