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. $\endgroup$ –
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 …
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.
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 scalar multiplication over ...
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 ...
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).
Associvity and Distributive property of Matrix Multiplication
It will follow from these facts that matrix multiplication is "obviously" associative. You may like to try to prove that above facts. For facts 2 and 3, the key concept is that linear maps are determined by the behavior on a basis, and this behavior is then encoded into a matrix (the columns of the matrix are the coordinates of the image of ...