Open sets preserved in linear transformation that isn't bijective?

[dumbQuestion]

Hi, I'm not sure how else to phrase this.Let's say I have a linear transformation from R³ to R². Let's assume in both spaces, I am using the standard topology with the standard euclidean distance metric. Does this mean that open sets in R³ will be mapped to open sets in R² under this transformation? What if the transformation is not one to one or onto?

If this is the case, I am not asking anyone to prove this to me but rather if you have any ideas what theorems I might look at to prove this myself? I was thinking connectedness would come into play. Am I right in that assumption?Thanks so much!

 

[micromass]

You should take a look at the open mapping theorem: http://en.wikipedia.org/wiki/Open_mapping_theorem_(functional_analysis)

 

[dumbQuestion]

Oh wow! Thank you for the quick reply! This is EXACTLY what I'm looking for!

 

[dumbQuestion]

I have one other question. Do you know if we can extend this thought? Say, the only way to map to an open set is from an open set? I'm not sure how to phrase it. I guess what I'm thinking is say, the pre-image of an open set, must is be an open set? But that would mean our transformation needs to have an inverse right?Here's my situation, maybe that will make my question clear. I keep coming across two scenarios in my research. Sometimes I am dealing with open sets in R^3 and I am mapping them to R^2 using a linear transformation. I wanted to know that the image of these open sets was an open set in R^2 and the theorem you provided seems to confirm that. But sometimes I am dealing with these unbounded regions in R^3 (for example I'll have something like a region that includes all points that lie above a given plane). I am still mapping these regions using the same linear transformation as before and I want to know that these unbounded regions map to unbounded region of R^2. I mean that's what makes sense to me and it seems like it should be obvious but I don't want to make any assumptions!EDIT: it just dawned on me, is it ethical to ask for help on an online forum regarding my research? I am not sure if I'm "allowed" to do that? I am really new to research so I'm not sure about that

 

[HallsofIvy]

I
EDIT: it just dawned on me, is it ethical to ask for help on an online forum regarding my research? I am not sure if I'm "allowed" to do that? I am really new to research so I'm not sure about that

The only way to answer that is to ask the people who have assigned you this research. I presume they gave it to you in order that you learn how to do research, and searching the internet is a form of research but they may want you to learn specific kinds of research in a specific order. Just as a teacher may ask a student to use the quadratic formula that could, in fact, be solved more easily by factoring, just to practice using the quadratic formula, so your teacher(s) may want you to focus first on searching books and journals, leaving the internet to later. But the only way to know is to ask them.

 

[lavinia]

Hi, I'm not sure how else to phrase this.Let's say I have a linear transformation from R³ to R². Let's assume in both spaces, I am using the standard topology with the standard euclidean distance metric. Does this mean that open sets in R³ will be mapped to open sets in R² under this transformation? What if the transformation is not one to one or onto?

If this is the case, I am not asking anyone to prove this to me but rather if you have any ideas what theorems I might look at to prove this myself? I was thinking connectedness would come into play. Am I right in that assumption?Thanks so much!

The linear map must be of maximal rank. The proof is elementary. In general if m < n then a linear map fron n space to m space is open if it is of maximal rank. If m > n, then it is not open.