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Math Help - Bases and Dimension....

  1. #1
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    Post Bases and Dimension....

    Hello can you please help me with this bases and dimension based problems:

    Q1) Give three different bases for F2 and for M[2x2] (F).

    Q2) Let U and V be distinct vectors of a vector space V. Show that if {u,v} is a basis for V and a and b are nonzero scalars, then both {u+v, au} and {au, bv} are also bases for V.

    Thanks a lot for the help!
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  2. #2
    MHF Contributor kalagota's Avatar
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    Quote Originally Posted by Vedicmaths View Post
    Hello can you please help me with this bases and dimension based problems:

    Q2) Let U and V be distinct vectors of a vector space V. Show that if {u,v} is a basis for V and a and b are nonzero scalars, then both {u+v, au} and {au, bv} are also bases for V.

    Thanks a lot for the help!
    1. it should be easy to show that \{u+v, au\} is a linearly independent set. as for the other criterion, it is enough to show that you can express u and v in terms of u+v and au: u = 0(u+v)+ a^{-1}au and v = 1(u+v) - a^{-1}au.

    you can do similar arguments for the other set..
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