Final answer:
The dot products uᵀv and vᵀu of two vectors in Rⁿ are equal due to the commutative property. The outer products uvᵀ and vuᵀ result in matrices that are transposes of one another.
Step-by-step explanation:
In the context of linear algebra within mathematics, the question is asking about the relationship between different products of two vectors: u and v from Rⁿ.
When we talk about uᵀv and vᵀu, we are referring to the dot product (also known as scalar product) of the two vectors, which in this case will be equal because the dot product is commutative. Specifically, uᵀv = vᵀu as each product sums up the products of the corresponding entries of the vectors.
Now, when we look at the products uvᵀ and vuᵀ, these are outer products, resulting in matrices rather than scalars. These matrices are not equal, but they are transposes of each other. Therefore, (uvᵀ)ᵀ = vuᵀ.