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Just trying to think if this is possible. What would a good example be if so?

To be a little more clear, if I have a linearly independent set of vectors $x_1, x_2, \ldots, x_k$, is there a linear mapping that will produce a linearly dependent set of output vectors?

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    $\begingroup$ Do you mean $\{A\vec{x}\}$ is linearly dependent? If so, the zero map would be a good example. If the preimage is a basis then any noninvertible map will do. $\endgroup$
    – anon
    Commented Oct 26, 2011 at 0:38

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If I am interpreting your question correctly, the answer is trivially yes. For example, the map that sends everything to $0$ is a linear map that will take any collection of vectors (linearly independent or not) to a set of vectors that is linearly dependent for trivial reasons.

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Note that if $T:V\to W$ is linear, and if the dimension of $W$ is less than $k$, then it is guaranteed that $\lbrace\,T(x_1),\dots,T(x_k)\,\rbrace$ will be a linearly dependent set. Heck, even if $T$ isn't linear.

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