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Math Help - Vector spaces Problem

  1. #1
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    Question Vector spaces Problem

    Let V1 = span {(1 0 2)}, and V2 = span{(0 0 1)} (they are colomn, not row)

    Observe that V1 and V2 are both vector spaces. A new set of vectors, S, is constructed by taking any vector from V1 and any vector from V2 and adding these 2 vectors together.

    why S will also be a vector space?
    How to find a basis for S and the dimension of S?
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  2. #2
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    In other words, S= \{w |w= u+ v, u\in V_1, u\in V_2\}.

    To show that a subset of a vector space is a subspace, you only need to show that the set is closed under addition and scalar multiplication of vectors. That is that if w_1 and w_2 are in S, so are w_1+ w_2 and \alpha w_1 where \alpha is any scalar.

    That can be done most conveniently, in one step, by showing that \alpha w_1+ w_2 is in the same set.

    Well, if w_1 is in that set then w_1= u_1+ v_1 for some v_1\in V_1 and some v_1\in V_2. If w_2 is in that set, similarly, w_2= u_2+ v_2.

    Now, \alpha w_1+ w_2= \alpha(u_1+ v_1)+ (u_2+ w_2)= (\alpha u_1+ u_2)+ (\alpha v_1+ v2). Do you see why that is again in S?

    Any vector in S is of the form w= u+ v. Since u is in the span of (1, 0, 2), u= \alpha(1, 0, 2) and since v is in the span of (0, 0, 1), v= \beta(0, 0, 1). That means that any vector in S can be written \alpha(1, 0, 2)+ \beta(0, 0, 1). You should be able to see from that what the dimension is and a basis.
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