我嘗試用我僅有既數學知識答以上幾個問題
一開頭有人問 點解Matrix addition is commutative, while Matrix multiplication is but not commutative.
可以從幾方面討論:
追求數學嚴謹性:
一個簡單既proof 就可以證明加法係commutative + 一個簡單既example 可以證明multiplication唔係commutative.
大學入面year 2/3學abstract algebra一定有問過點解matrix係ring but not a commutative ring.(其實接近係同一條問題, 只係presentation唔同)
以上既理據本身都足夠去回答呢個問題=數學上一個counter example已經足夠證明一樣野係錯
1966年有一篇paper都係用短短一頁紙寫一個counter example證明一件事係錯.
追求數學既motivation/歷史:
數學上matrix (根據wiki)係 motivated by representations of linear map
如果你答"因為composition of linear maps are not necessarily commutative"
咁你就要問啦: 呢樣野係咪well-known/係你既課程上係咪standard knowledge, 係既, 呢個係一個答案
唔係既, 你就要證明呢件事, 而證明呢件事你可以
1."因為linear map其實都係equivalent to 另一樣野->因為佢唔commutative 所以linear map composition都唔commutative"
->呢個exercise其實係將一個Definition同另一個definition同等左佢, 去到最最後你接受到/學到邊一倨definition係唔Commutative.
2."你證明linear map composition係唔commutative"
technically呢個唔係一個簡單問題(對中學生黎講)
linear maps are functions from a vector space to another vector space that satisfy f(ax)=af(x) and f(x) + f(y)
第一咩係vector space, 第二vector space can be infinite-dimensional
你證明呢個命題其實遠比你需要既野目標廣闊.
加上你證明完呢樣野, 你都要證明番, linear map between finite-dimensional vector spaces等價於matrix, 而係呢個case你既non-commutativity又係咪valid
(因為數學上有時個命題係general case成立, special case唔成立
e.g: matrix multiplication in 1x1 dimension is commutative.)
實用方面:
回答番, 咁如果我想用呢個切入點(matrix其實係一種"特別既"function=linear map)
然後話因為f(g(x))=/=g(f(x)) in general.
我反而覺得咁樣太過general
正正因為數學上有時真係special case會成立, 如果有D叻既學生走黎Challenge你你反而答唔到佢點解, 最後可能都係用番一個formal proof去處理, 即係前面提過proof of addition+A counter-example of multiplication.