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"Rasmus Rosendahl-Kaa (S255955)", + semester: "2025 Fall", +) + +#set math.vec(delim: "[") +#set math.mat(delim: "[") +#set text(lang: "da") + + += Matrix +$underline(underline(A))$ - en matrix\ +$underline(v)$ - en vektor + +Har en matrix: +$ +underline(underline(A)) = mat(a_11,a_12,dots,a_(1n);dots.v,,,dots.v;a_(m 1),a_(m 2),dots,a_(m n)) space in FF ^(m times n) +$ + +*Søjlevektor:*\ +$ mat(a_11;dots.v;a_(m 1)) in FF^(m times 1) = FF^(m) $ +*Rækkevektor:*\ +$ mat(a_11,a_12,dots,a_(1 n)) in FF^(1 times n) $ + +#definition(title: "Addition af vektorer")[ +To vektorer $underline(a) "og" underline(b)$: $underline(a)=mat(a_1;dots.v;a_m), underline(b)=mat(b_1;dots.v;b_m)$ + +$ underline(a)+underline(b)=mat(a_1+b_1;dots.v;a_m+b_m) $ +] + +#definition(title: "Skalar af vektor")[ +Har en vektor $underline(a)$: $underline(a)=mat(a_1;dots.v;a_m)$ og en skalar $c in FF$: + +$ c dot underline(a) = mat(a_1 dot c;dots.v;a_m dot c) $ +] + +#definition(title: "Lineær kombination af vektor")[ +$underline(v_1),underline(v_2),dots,underline(v_n) in FF^m "og" c_1,c_2,dots,c_n in FF$ +$ c_1 dot underline(v_1) + c_2 dot underline(v_2) + dots + c_n dot underline(v_n) $ +Hedder en lineær kombination +] + += Lineær uafhængighed +#definition(title: "7.1.1: Lineær uafhængighed")[ +Et sæt vektorer $(underline(v_1),underline(v_2),dots,underline(v_n)$ kaldes lineær uafhængige hvis og kun hvis ligningen: +$ c_1 dot underline(v_1) + c_2 dot underline(v_2) + dots + c_n dot underline(v_n) $ +kun har én løsning: $c_1=c_2=dots=c_n=0$ +] diff --git a/01001 - Matematik 1a/Noter/Polynomier og induktion/Polynomier og induktion lille dag 03-10-2025.pdf b/01001 - Matematik 1a/Noter/Polynomier og induktion/Polynomier og induktion lille dag 03-10-2025.pdf new file mode 100644 index 0000000..92e7c36 Binary files /dev/null and b/01001 - Matematik 1a/Noter/Polynomier og induktion/Polynomier og induktion lille dag 03-10-2025.pdf differ diff --git a/01001 - Matematik 1a/Noter/Udspænding, ordnet basis og 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Cantor-Schröder-Bernstein theorem/Graphs and the Cantor-Schröder-Bernstein theorem.pdf +++ b/Diskret Mat/Graphs and the Cantor-Schröder-Bernstein theorem/Graphs and the Cantor-Schröder-Bernstein theorem.pdf @@ -11133,8 +11133,8 @@ endobj 250 0 obj << /Creator (Typst 0.14.0) - /ModDate (D:20251127102253Z) - /CreationDate (D:20251127102253Z) + /ModDate (D:20251128064704Z) + /CreationDate (D:20251128064704Z) >> endobj @@ -11145,7 +11145,7 @@ endobj /Subtype /XML >> stream -Typst 0.14.0en2025-11-27T10:22:53+00:002025-11-27T10:22:53+00:008application/pdfjQ9JEaMS+nVSciNI1uRlNw==jQ9JEaMS+nVSciNI1uRlNw==proof1.7 +Typst 0.14.0en2025-11-28T06:47:04+00:002025-11-28T06:47:04+00:008application/pdfpFgGGEBr1VEaqg0mHBbt3w==pFgGGEBr1VEaqg0mHBbt3w==proof1.7 endstream endobj @@ -11427,7 +11427,7 @@ trailer /Size 253 /Root 252 0 R /Info 250 0 R - /ID [(jQ9JEaMS+nVSciNI1uRlNw==) (jQ9JEaMS+nVSciNI1uRlNw==)] + /ID [(pFgGGEBr1VEaqg0mHBbt3w==) (pFgGGEBr1VEaqg0mHBbt3w==)] >> startxref 3375983 diff --git 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