dot products and duality in linear algebra
The dot product is fundamentally defined by vector duality in linear algebra, establishing a unique correspondence between vectors and linear transformations from $n$-dimensional space to the scalar field $\mathbb{R}$. Theoretically, applying such a transformation (projection onto a subspace) followed by scaling along that projection's direction is structurally equivalent to taking the dot product of two vectors. This concept resides within vector algebra and serves as a primary mechanism for relating geometric entities (vectors/arrows) to functional spaces (transformations/matrices).
dot products and duality in linear algebra
The dot product is fundamentally defined by vector duality in linear algebra, establishing a unique correspondence between vectors and linear transformations from $n$-dimensional space to the scalar …