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Math Help - Show a set is not a subspace of R^4

  1. #1
    Member
    Joined
    Jan 2011
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    Show a set is not a subspace of R^4

    Hi all,

    Having a problem trying to show that the set <br />
\begin{bmatrix} 3a-b\\ ab \\ a+b\\ b+2a \end{bmatrix}<br />
    where a and b are real numbers, is not a subspace in R^4..
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  2. #2
    Senior Member
    Joined
    Feb 2008
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    Re: Show a set is not a subspace of R^4

    Let a=1,b=2 and consider the vector

    2\left[\begin{array}{c}3a-b\\ab\\a+b\\2a+b\end{array}\right]=2\left[\begin{array}{c}1\\2\\3\\4\end{array}\right]=\left[\begin{array}{c}2\\4\\6\\8\end{array}\right].

    Then 3a'-b'=2, a'b'=4 and a'+b'=6. But this system has no solution:

    3a'-b'=2 implies b'=3a'-2. Then a'+b'=6 gives us a'+(3a'-2)=6 and therefore a'=2 and b'=4. But then a'b'=8, contradicting the condition a'b'=4.
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