Control

Mathematics for Business I
1st G.A.D.E., Academic Year 2011/12 Control Unit 1, option B
SURNAME(S):.............................................................. NAME: ......................................

→ → → 1 → 1. In IR3 , consider the vectors −1 = (−2, 2, 1), −2 = (−1, 1, 2 ), −3 = (0, −1,0), −4 = u u u u (3, 3, 3)}. Answer the next questions, explaining the reasoning you use for your answers. → → → → a) Study if the vectors −1 , −2 , −3 ,−4 are linearly dependent or indepenu u u u dent. − → → → b) Study if the vectors →, −2 , −3 , −4 are a generating system in IR3 . u1 u u u − , →, − , − abasis in IR3 ? If not, use those vectors → − → → c) Are the vectors u1 u2 u3 u4 to build up a basis in IR3 (that is to say, using those vectors, addand/or drop some vectors to build up a basis in IR3 ). → → d ) Let S be a vector subspace with basis {−1 , −3 }. Calculate dim(S), u u the equation(s) of Sand another vector in S. − e) Let S be a vector subspace with basis {→}. Calculate dim(S), the u3 equation(s) of S and another vector in S. 2. Let’sconsider the maps (1) f : IR4 −→ IR2 such as f (x, y, z, t) = (x + 2y + 3t, (2) f : IR4 −→ IR2 such as f (x, y, z, t) = (x + 2y + 3t,
1 2 1 2

x − z − 10)x − z).

a) Say which one is not a linear map and why it isn’t. For the map that it is linear, answer the next questions: b) Calculate its associatedmatrix. → c) Calculate the image of the vector − = (1, −1, 2, 0). u − → → → d ) Calculate one vector − = (x, y, z, t) verifying f (− ) = 0 v v

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