linear algebra - Proving Distributivity of Matrix Multiplication ...
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:
elementary number theory - Why is the distributive law an axiom ...
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. $\endgroup$ –
"implied multiplication" operator precedence? - Mathematics …
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 prove distributive property of a determinant?
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.
Distributive property of matrix-vector multiplication?
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
Distributive modulo? - Mathematics Stack Exchange
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 …
Prove that real multiplication distributes over addition
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?
Why exactly does the distributive property work?
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.
linear algebra - Distributive Property Of Matrix Multiplication ...
2018年1月3日 · The algebraic structure formed by matrices under addition and multiplication is called a ring. Rings have distributive properties. If you want to explore structures with addition and multiplication but without distributive properties, perhaps check out Ring without distributive law? and Example of "ring" without the distributive property?
abstract algebra - Multiplication of negative numbers explanation ...
2024年10月1日 · $\begingroup$ As I explained in an answer there, every nondegenerate theorem of rings necessarily uses the distributive law since it is the only ring law connecting the additive and multiplicative structure (here nondegenerate theorem means the theorem doesn't belong to the underlying additive group or multiplicative monoid).