Without a bit of information about what files are in your repository (pure source code, images, executables, ...), it's a bit hard to answer the question :)
Beside this, I'll consider that you're willing to default to LF as line endings in your working directory because you're willing to make sure that text files have LF line endings in your .git repository wether you work on Windows or Linux. Indeed better safe than sorry....
However, there's a better alternative: Benefit from LF line endings in your Linux workdir, CRLF line endings in your Windows workdir AND LF line endings in your repository.
As you're partially working on Linux and Windows, make sure core.eol is set to native and core.autocrlf is set to true.
Then, replace the content of your .gitattributes file with the following
* text=auto
This will let Git handle the automagic line endings conversion for you, on commits and checkouts. Binary files won't be altered, files detected as being text files will see the line endings converted on the fly.
However, as you know the content of your repository, you may give Git a hand and help him detect text files from binary files.
Provided you work on a C based image processing project, replace the content of your .gitattributes file with the following
* text=auto
*.txt text
*.c text
*.h text
*.jpg binary
This will make sure files which extension is c, h, or txt will be stored with LF line endings in your repo and will have native line endings in the working directory. Jpeg files won't be touched. All of the others will be benefit from the same automagic filtering as seen above.
In order to get a get a deeper understanding of the inner details of all this, I'd suggest you to dive into this very good post "Mind the end of your line" from Tim Clem, a Githubber.
As a real world example, you can also peek at this commit where those changes to a .gitattributes file are demonstrated.
UPDATE to the answer considering the following comment
I actually don't want CRLF in my Windows directories, because my Linux environment is actually a VirtualBox sharing the Windows directory
Makes sense. Thanks for the clarification. In this specific context, the .gitattributes file by itself won't be enough.
Run the following commands against your repository
$ git config core.eol lf
$ git config core.autocrlf input
As your repository is shared between your Linux and Windows environment, this will update the local config file for both environment. core.eol will make sure text files bear LF line endings on checkouts. core.autocrlf will ensure potential CRLF in text files (resulting from a copy/paste operation for instance) will be converted to LF in your repository.
Optionally, you can help Git distinguish what is a text file by creating a .gitattributes file containing something similar to the following:
# Autodetect text files
* text=auto
# ...Unless the name matches the following
# overriding patterns
# Definitively text files
*.txt text
*.c text
*.h text
# Ensure those won't be messed up with
*.jpg binary
*.data binary
If you decided to create a .gitattributes file, commit it.
Lastly, ensure git status mentions "nothing to commit (working directory clean)", then perform the following operation
$ git checkout-index --force --all
This will recreate your files in your working directory, taking into account your config changes and the .gitattributes file and replacing any potential overlooked CRLF in your text files.
Once this is done, every text file in your working directory WILL bear LF line endings and git status should still consider the workdir as clean.