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I have a $T\colon\mathbb{R}^2\to\mathbb{R}^2$ linear operator given by $$T(x,y) = (2x - y, -8x + 4y)$$

How can I tell if the vector $(1, -4)$ is in $R(T)$?

Ok so I set everything into a matrix:

$$\left( \begin{matrix} 2 & -1 & 1\\ -8 & 4 & -4\\ \end{matrix}\right) $$

I row reduced it and found that it was linearly dependent. So I'm going to assume that that means it's is in R(T).

Edit: I'm not sure if I did that right because it isn't set to 0.

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    What linear algebra do you know so far?2011-04-21
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    He's not making fun of you; he's trying to find out how much linear algebra you know in order to know how to answer your question. There are "low tech" and "high tech" ways of solving this problem, depending on how much linear algebra (and at what level) you know. For example, one can answer by invoking the Dimension Theorem to deduce that the range has dimension $1$, and then easily deduce whether the vector $(1,-4)$ is in the that one dimensional subspace. Or it can be done by solving a system of linear equations. He's trying to help at an appropriate level.2011-04-21
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    I was only kidding about the "making fun of me" part, sorry. I'm currently trying to understand Kernel and Ranges for linear operators.2011-04-21
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    I am adding this comment here (for now) because you deleted the other question. If what you have is a problem taken from a book/assignemnt, it is perfectly fine to *quote it* (say, in a quote box by preceding the first character in the paragraph with `> `, and then write down your own comments and describe your doubts. If you are confused about what a problem is asking you to do, then that's the way to go. But your recently deleted question was an absolute mess: it started halfway through, and you never even said what "the problem" you were trying to solve *was*.2011-04-26

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