Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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?

share|improve this question
4  
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. –  anon Oct 26 '11 at 0:38

2 Answers 2

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.

share|improve this answer

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.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.