By Stanley I. Grossman ; traductor: Alfonso Leal Guajardo ; revisor tecnico: Francisco Paniagua Bocanegra.

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An X n |αi ∈ R, i = 0, n, n ∈ N∗ be the set of polynomials in the indeterminate X, with real coefficients, of degree ≤ n. (i) Show that the set Rn [X] is a real vector space, denoted by (Rn [X], +, ·) with the usual operations of addition of polynomials and of multiplication of the polynomials with scalars from R. (ii) Find the dimension of Rn [X]. 30 1 Vector Spaces Solution (i) Let be α ∈ R, P, Q ∈ Rn [X], P (X) = a0 + a1 X + . . + an X n , Q (X) = b0 + b1 X + . . + bn X n . Stage I. 11).

It results that − → − → OA × OB AπOAB = 2 . 35 (see [9], p. 52). Knowing two sides AB = 3i − 4j BC = i + 5j of a triangle, calculate the length of its height CD. Solution AB = √ 9 + 16 = 5 i j k AB × BC = 3 −4 0 = 19k 1 5 0 AπABC = AB × BC AB CD AB × BC 19 = = =⇒ CD = . 36 (see [5], p. 65). Let the vectors a = 3m − n b = m + 3n be such that m = 3 and n = 2 and ≺ (m, n) = π2 . Determine the area of the triangle formed by the vectors a and b. 34, the area of the triangle formed by the vectors a and b is: 1 Aπ = a×b .

If the free vectors a, b, c ∈ V3 \ 0 are noncoplanar, then the volume of the parallelepiped determined by these three vectors (see Fig. 15) is given by a · b × c . Fig. e. a· b×c = Vparalelipiped . 39 (see [2], p. 19 and [1], p. 121). The mixed product of the free vectors has the following properties: 1. a · b × c = c · a × b = b · (c × a) 2. a · b × c = −a · c × b 3. ta · b × c = a · tb × c = a · b × tc , (∀) t ∈ R; 4. a + b · c × d = a · c × d + b · c × d 5. Lagrange identity: a×b · c×d = a·c a·d b·c b·d 6.