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Linear Algebra Field & Vector Space Problem

  1. Sep 17, 2013 #1
    1. The problem statement, all variables and given/known data
    Let V1 and V2 be vector spaces over the same field F.
    Let V = V1 X V2 = {f(v1, v2) : v1 [itex]\in[/itex] V1; v2 [itex]\in[/itex] V2}, and de fine addition and scalar multiplication as follows.
     For (v1, v2) and (u1, u2) elements of V , defi ne (v1, v2) + (u1, u2) = (v1 + u1, v2 + u2).
     For (v1, v2) element of V and c [itex]\in[/itex] F, defi ne c  (v1, v2) = (c  v1, c  v2).
    a) In the de finitions of addition and scalar multiplication there are three "+" and three "." To
    which vector space is each one associated with?
    b) Show that V is a vector space. NB: you must provide some reason why each of the axioms is
    satis ed.


    2. Relevant equations

    To be absolute honest i have no idea what it means when it asked which vector space it belong to in part a).
    ANd for part 2, i do not know where to start.


    3. The attempt at a solution

    I know that to proof fields or vector spaces, it has to satisfy with the axioms

    Zero vector
    addition
    scalar multiplication
    and etc.
    Just have trouble starting this problem

    Thanks.

    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Sep 17, 2013 #2

    vela

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    Take, for example, ##\vec{x}+\vec{y}## where ##\vec{x},\vec{y} \in \mathbb{R}^2##. You write ##\vec{x} = \begin{bmatrix} x_1 \\ x_2 \end{bmatrix}## and ##\vec{y} = \begin{bmatrix} y_1 \\ y_2 \end{bmatrix}##. Then
    $$\vec{x}+\vec{y} = \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} + \begin{bmatrix} y_1 \\ y_2 \end{bmatrix} = \begin{bmatrix} x_1+y_1 \\ x_2+y_2 \end{bmatrix}.$$ The plus sign between ##\vec{x}## and ##\vec{y}## represents a different operation than the plus sign between ##x_1## and ##y_1##. Why? Because the first one is about adding two vectors while the second one is about adding two real numbers.
     
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