So, matrix multiplication is just the image of composition of linear transformations under the identification of matrices with linear transformations. In particular, then, distributivity of matrix multiplication is really just distributivity of composition of linear transformations, which lends itself to a far more transparent proof:

2021年10月17日 · $\begingroup$ If multiplication is how we find "areas" then the distributive property assures us that we can find "areas" by creating subdivisions and summing their "areas." This is the intuition behind the distributive property, and probably why it was first recognized as necessary. $\endgroup$ –

2021年3月7日 · Distributive Property Of Matrix Multiplication 0 Matrix-vector product: representing matrix as vector of vectors seemingly leads to paradox when transposing the matrix

2021年4月26日 · Why was this property accepted as an axiom? It doesn't seem trivial to me like other axioms of real numbers. Was the multiplication between two negative real numbers or a negative real number & a positive real number defined before setting up the distributive property of real numbers as an axiomatic property of real numbers?

The OP seems to be using modulo as an operator; and that operator is NOT distributive. If we let $\%$ be the the 'modulus' operator (as it is usually denoted in computer languages) then $$(2 + 2) \% 3 = 1 \neq (2 \%3) + (2 \% 3) = 2 + 2 = 4$$ In mathematical discourse, modulus is …

2014年11月18日 · How to prove that $|A\\cdot B| = |A|\\cdot|B|$ where A and B are square matrices of the same size? P.S.: This proof is not mentioned in my textbook, nor was I able to find it on the web.

2015年7月28日 · I need help with a simple proof for the distributive property of scalar multiplication over scalar addition. Help with proving this definition: $(r + s) X = rX + rY$ I have to prove the truth of ...

2014年9月15日 · You are left with a true statement. a(b+c) will always equal a(b+c) because something will always equal itself. If a(b+c)=ab+bc is the distributive property, and I just simplified the distributive property down to a true statement, then the distributive property must equal the true statement because simplifying something does not change its value.

2019年5月19日 · Distributive law, in mathematics, the law relating the operations of multiplication and addition, stated symbolically, a(b + c) = ab + ac; that is, the monomial factor a is distributed, or separately applied, to each term of the binomial factor b + c, resulting in the product ab + ac. –Encyclopedia Britannica

How to show that multiplication of matrices, when defined, is associative and distributive over addition ...