Linearna algebra, pismeni ispit - Univerzitet u Zenici · 2016-03-22 · Univerzitet u Zenici...
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Univerzitet u ZeniciFilozofski fakultetOdsjek: Matematika i informatikaZenica, 01.09.2014.
Linearna algebra, pismeni ispit
1. U vektorskom prostoru P4 realnih polinoma stepena ≤ 4 dat je skup
M = {p ∈ P4 | p′(0) = p(1), p′′(0) = 2p(−1)}.
Dokazite da je M vektorski potprostor prostora P4, odredite mu jednu bazu i dimenziju, te jedandirektni komplement.
2. Neka je [2 11 4
]matrica linearnog operatora T : V2(0)→ V2(0) u kanonskoj bazi
{~i =
(10
),~j =
(01
)}. Odrediti
matricu operatora T u bazi {~i + 2~j,~i + 3~j}. Da li postoji vektor ~v ∈ V2(0) takav da jeT (~v) = 3~i + 5~j?
3. (20%)(a) Objasniti sta ce se desiti kada se Gram-Schmidtov proces primjeni na linearnozavisan skup vektora.
(80%)(b)Zadan je unitarni prostor Mat2×2(R) sa skalarnim (unutrasnjim) proizvodom〈A,B〉 = trag(A>B) i neka je L vektorski potprostor Mat2×2(R) definiran kao
L = span
{(0 33 −3
),
(−2 −26 −2
),
(−1 1−1 1
)}.
Nadite ortonormiranu bazu za L.
4. Data je matrica
A =
1 a 1 01 −1 0 10 0 1 01 b 0 1
.
Odrediti parametre a i b ako je poznato da je A singularna matrica cije sve svojstvene vrijednostiimaju algebarsku visestrukost 2.
Vazno: Ovaj papir treba predati zajedno s rjesenjima zadataka! Svaku formulu koju mislitekoristit, u sva 4 zadatka, obavezno napisati, kao i znacenja simbola iz formule. Ispit pisati iskljucivohemiskom olovkom plave ili crne tinte. Prije rjesenja prepisati postavku (tekst) zadatka.
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Zadaci su skinuti sa stranice ff.unze.ba/nabokov.Za uocene greske pisati na [email protected]
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