From e5b62bfaf1c796f5dfbbb973da0cf5e06bb0fb34 Mon Sep 17 00:00:00 2001 From: Rasmus Rosendahl-Kaa Date: Wed, 26 Nov 2025 15:38:41 +0100 Subject: [PATCH] =?UTF-8?q?samme=20som=20f=C3=B8r?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- .../Hjemmeaflevering nr. 1/main.typ | 87 ++++++++++++++++++ 01001 - Matematik 1a/test/test.pdf | Bin 8671 -> 0 bytes 01001 - Matematik 1a/test/test.typ | 8 -- 3 files changed, 87 insertions(+), 8 deletions(-) create mode 100644 01001 - Matematik 1a/Hjemmeaflevering nr. 1/main.typ delete mode 100644 01001 - Matematik 1a/test/test.pdf delete mode 100644 01001 - Matematik 1a/test/test.typ diff --git a/01001 - Matematik 1a/Hjemmeaflevering nr. 1/main.typ b/01001 - Matematik 1a/Hjemmeaflevering nr. 1/main.typ new file mode 100644 index 0000000..6b5911b --- /dev/null +++ b/01001 - Matematik 1a/Hjemmeaflevering nr. 1/main.typ @@ -0,0 +1,87 @@ + +#set math.vec(delim: "[") +#set math.mat(delim: "[") +#set text(lang: "da") + += Problem A +Lad $W$ være udspændt af følgende vektorer i $RR^3$: +$ + bold(v)_1 = vec(-1,1,0), quad bold(v)_2 = vec(5,4,3), quad bold(v)_3 = vec(7,11,6) +$ + +Angiv en ordnet basis for $W$ + +== Løsning +Laver en totalmatrix + += Problem B +Lad $C_infinity (RR)$ være det reelle vektorrum fra Eksempel 10.4.5 i lærebogen. Der defineres en +funktion $L: C_infinity (RR) arrow C_infinity (RR)$ ved $L(f) = f' +f-1$ hvor udtrykket $f'$ betegner den afledte funktion af $f$. Er $L$ en lineær afbildning? + + + + + += Problem C +Lad $F: CC^2 arrow CC^2$ være defineret som følger: +$ + F(vec(v_1,v_2)) = mat(1,1;-4,5) dot vec(v_1,v_2), quad v_1v_2 in CC +$ + +Der gives ordnede baser +$ + beta = (vec(1,2),vec(-2,1)) "og" gamma = (vec(1,1), vec(0,1)) op("for") CC^2 +$ + +Beregn afbildningsmatricen $mat(F,beta,gamma)$. + + + + + += Problem D +Der vælges følgende ordnede basis for det reelle vektorrum $RR^(2 times 2)$: +$ + beta = (mat(1,0;0,0), mat(0,1;0,0), mat(0,0;1,0), mat(0,0;0,1)) +$ + +Givet den lineære afbildning $M: RR^(2 times 2) arrow RR^(2 times 2)$ defineret ved +$ + M(bold(A)) = mat(1,2;-1,-2) dot bold(A), quad bold(A) in RR^(2 times 2) +$ + +Beregn afbildningsmatricen $mat(M,beta,beta)$. + + + + + += Problem E +Givet følgende matrix +$ + mat(2,0,0;2,1,-1;2,-1,1) in RR^(3 times 3) +$ + +Bestem matricens egenværdier samt ordnede baser for de tilhørende egenrum. + + + + + += Problem F +Om et inhomogent lineært ligningssystem over $RR$ med fire ligninger og to ubekendte oplyses +at +$ + bold(v)_p = vec(1,-1) in RR^2 +$ +er en partikulær løsning. Er vektoren $3 dot bold(V)_p$ en løsning til systemet? + + + + + += Problem G +lad $V$ være det reelle vektorrum $RR^(3 times 3)$. 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