Hvad er forskellen mellem base, afledt, skalær og vektormængde?


Svar 1:

Du forvirrer muligvis fysik med matematik:

En basismængde er en fysisk mængde med fysiske egenskaber som masse, afstand, ladning, tid osv.

En afledt mængde er en fysisk kombination deraf, såsom m / s, m ^ 3 / kg osv.

En skalar er en matematisk 1-d-mængde (selvom tal naturligvis ikke kun er reelle tal, men også 2-d-komplekse tal).

En vektor er en matematisk enhed med 2 eller flere skalære komponenter, der normalt er ortogonale.

FORVIS IKKE matematiske dimensioner (2-d) med fysiske basismængder.

Eksempel: Matematik er 2 + 3 = 5; Fysik er 2 kg + 3 kg = 5 kg;


Svar 2:

ScalarisandelementofthefieldFdefinedbelow,usuallyjustarealnumbervaluedvariable.Thinkofitasanumberthatindicatesthescaleorhighbigsomethingis.Itcouldalsobenegative.Abaseinlinearalgebraisawayofpartitioningavectorspacewhereyoucangeneratealloftheelementsofthevectorspaceusingthisbaseandtheadditionalpropertythatisimportantisthattheelementsofyourbasebelinearlyindependent(youcannotcombinetheotherbaseelementsinalinearfashiontoobtainanyoftheotherbaseelements).Theformaldefinitionwouldbeasetofvectors[math]v1,v2,v3...V[/math]suchthatforany[math]wV[/math]thereisasetofscalars[math]a1,a2,a3,...F[/math]suchthat[math]w=a1v1+a2v2+a3v4+...[/math]withtheadditionalpropertythattheequation[math]via1v1+a2v2...[/math]alwaysholdsandwhere[math]vi[/math]isnotpartofthesumontherighthandside.Avectorquantityissimplyanelementofavectorspace.Youwouldneedtoknowthedefinitionofavectorspacewhichis:Scalar is and element of the field \mathbb{F} defined below, usually just a real number valued variable. Think of it as a number that indicates the scale or high big something is. It could also be negative. A base in linear algebra is a way of partitioning a vector space where you can generate all of the elements of the vector space using this base and the additional property that is important is that the elements of your base be linearly independent (you cannot combine the other base elements in a linear fashion to obtain any of the other base elements). The formal definition would be a set of vectors [math]v_1,v_2,v_3... \in \mathbb{V}[/math] such that for any [math]w \in \mathbb{V}[/math] there is a set of scalars [math]a_1,a_2,a_3,...\in \mathbb{F}[/math] such that [math]w = a_1\cdot v_1 + a_2\cdot v_2 + a_3 \cdot v_4 +...[/math] with the additional property that the equation [math]v_i \not= a_1\cdot v_1 + a_2\cdot v_2...[/math] always holds and where [math]v_i[/math] is not part of the sum on the right hand side. A vector quantity is simply an element of a vector space. You would need to know the definition of a vector space which is:

letVbeavectorspace.let \mathbb{V} be a vector space.

Forv,wV,youhavea[math]v+bwV[/math],where[math]a,bF[/math]where[math]F[/math]issomefield,usually[math]R[/math],therealnumbers.For v,w \in \mathbb{V}, you have a [math]\cdot v + b\cdot w \in \mathbb{V}[/math], where [math]a,b \in \mathbb{F}[/math] where [math]\mathbb{F}[/math] is some “field”, usually [math]\mathbb{R}[/math], the real numbers.

Inessence,thevectorsaresuchthatthevectorspaceVisclosedunderlinearcombinationsofitselements.In essence, the vectors are such that the vector space \mathbb{V} is closed under linear combinations of it’s elements.

Usually,intheearlierapplicationsthatstudentsseevectorspaces,youhavevectorsrepresentedasanangleandlength.SoifvV,[math]v=(r,θ)[/math],where[math]r[/math]isarealnumberedvaluedefiningthelengthand[math]θ[/math]isanangle.Usually, in the earlier applications that students see vector spaces, you have vectors represented as an angle and length. So if \overrightarrow{v} \in \mathbb{V}, [math]\overrightarrow{v} = (r,\theta)[/math], where [math]r[/math] is a real numbered value defining the length and [math]\theta[/math] is an angle.

Definitionen af ​​et felt er her wikipedia-siden

Felt (matematik) - Wikipedia

Men i de fleste tilfælde bruger de rigtige tal, medmindre du studerer mere avancerede emner.