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Math Help - Determing if F linear map

  1. #1
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    Determing if F linear map

    Suppose that F: \mathbb{R}^4 \to \mathbb{R}^2 is a function satisfying

    F(\boldsymbol{a}) = \begin{pmatrix}3\\-1\end{pmatrix}, F(\boldsymbol{b}) = \begin{pmatrix}3\\0\end{pmatrix}, F(\boldsymbol{u}) = \begin{pmatrix}1\\-3\end{pmatrix}, F(\boldsymbol{v}) = \begin{pmatrix}4\\1\end{pmatrix} and F(\boldsymbol{w}) = \begin{pmatrix}0\\4\end{pmatrix}

    Is F a linear map? Explain.

    Im not sure how to do this.
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  2. #2
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    What are a, b, u, v, and w? Have you got no information about these?
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  3. #3
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    Quote Originally Posted by HappyJoe View Post
    What are a, b, u, v, and w? Have you got no information about these?
    No info was given.
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  4. #4
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    Simply giving the results of F on 4 separate vectors is not enough to tell whether F is linear or not- it is not enough to determine what F is!

    For example, it would be possible to define F to be exactly what is given on those four vectors and 0 for all other vectors. That would NOT be a linear map.

    If we are given the result of applying F to four basis vectors, a, b, u, v, for R^4 and F is "defined by linearity" for all other vectors (that is, if a, b, u, v are basis vectors, then for any v in R^4, v= pa+ qb+ ru+ sv for some numbers p, q, r, and s, and then F(v)= pF(a)+ qF(b)+ rF(u)+ sF(v), then F is, of course, linear.
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