Final answer:
The question is about determining whether certain operations are valid inner products on an n-dimensional real space, with a focus on properties such as positivity, symmetry, linearity, and positive-definiteness.
Step-by-step explanation:
The question asks to validate whether certain operations define an inner product on a vector space, specifically ℝ^n (the n-dimensional real space). An operation is an inner product if it satisfies four conditions: (1) positivity, (2) symmetry, (3) linearity in the first argument, and (4) positive-definiteness. According to the provided snippets, some operations involve the scalar product or dot product (which is a valid inner product), whereas others involve vector operations like the cross product or operations not well-defined between vectors, such as division.
For example, C = A · B represents a scalar product, which is an inner product. However, operations like C = A × B or Č = Ã /× B do not define inner products because the cross product is not commutative or scalar, and division is not defined for vector spaces. To establish conformity with inner product properties, vector products and addition operations should be examined against the specified conditions, noting that scalar and vector product treat vectors differently.