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

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 …

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 …

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

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 …

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 …

Wikipedia also mentions it in the article on Matrix Multiplication, with an alternate name as the Schur product. As for the significance of element-wise multiplications (in signal processing), …

Dec 15, 2017 · You can invent your own product or way of multiplication, but the standard product of matrices only works, as you say, when the number of columns of the first matrix matches …

I cannot find a way to prove it You have an explicit formula (I'm assuming, of course, the naive matrix multiplication algorithm) with additions and multiplications in it, and you have a …

Feb 19, 2019 · Find the computational cost of a column vector $x$ multiplied by a row vector $v$ I computed n multiplication operations and n - 1 addition operations, so would that make for $n …