Aug 31, 2019 · One motivation for these types of vector multiplication is that we can perform geometric calculations in three-dimensions through purely algebraic methods; for example, …

Jan 2, 2023 · The vector dot and cross product are not some kind of analogue or generalization of the arithmetical product's sense of repeated addition. (Indeed, I try to avoid using "multiply", …

Long story short, the question is simple. Is matrix multiplication just a special case of the dot product of two sets of vectors when the sets of vectors have the same cardinality and all …

I recommend writing componentwise multiplication of vectors using some symbol that does not have a standard meaning, perhaps ⋆ ⋆ (\star) or ⋄ ⋄ (\diamond), so that people reading won't …

Apr 13, 2017 · In some mathematical topics (probability transition matrices for Markov chains) the convention is typically a row vector times a matrix. Both of these are just special cases of …

Mar 26, 2015 · If you add additional structure to the vector space by giving meaning to products of the form $\vec {v}a$, that's fine, but it's not part of the underlying vector space structure. It's a …

Feb 4, 2011 · On the other hand the dot product of two vectors gives the outcome of an operation applied on the vectors involved by considering the physics of the problem . Hence dot product …

Jul 22, 2015 · Scalar Multiplication Example: $–10 × (1, –7) = (–10 × 1, –10 × –7) = (–10, 70)$, where –10 is a scalar. Under these definitions for the operations, it can be rigorously proven …

Note, though, that a a is a column vector, but aT a T is a row vector. The dot product is only defined for two vectors of the same type, so your expressions aT ⋅ a a T ⋅ a and a ⋅aT a ⋅ a T …

I was just wondering why we don't ever define multiplication of vectors as individual component multiplication. That is, why doesn't anybody ever define $\\langle a_1,b_1 \\rangle \\cdot …