Depends on what you mean by "web software engineering." Really it's almost a useless term, since a web-frontend can be the front-end for almost any sort of system. The question really should just be "where does linear algebra have applications to software engineering?" in a general sense.

Pedantry & ranting aside, the obvious place is anything to do with machine learning / text mining. If processing large amounts of text and searching for things within, or based on, that text, you may well wind up using algorithms that involve linear algebra.

One of the Mahout project founders speaks fairly often in the Triangle area, at User Group meetings and such, and he always seems to mention Linear Algebra in his talks. There's a reason for that.

Google? The Page Rank algorithm is basically Linear Algebra.

ButI think you're asking the wrong question. You're doing the equivalent of a 14-year-old saying "When am I ever going to need this?" Sometimes the benefits are much more nebulous and intangible - they affect the way you think, and the way you approach problems.

Most programming doesn't use advanced math, but advanced math helps you do the programming.

(Note - all the above personal opinion, and I don't have specific studies and citations to back it up)

Making sense of large corpus of text via machine learning, say .. SVM which stands on top of linear algebra (computing inner products in an isomorphic space of bigger dimension via kernel functions) (Hillary Mason working at bit.ly gives talks about this stuff http://www.hilarymason.com/ )

Data mining, a lot of which boils down to multidimensional exploratory statistics needs linear algebra.

Advance memory allocation uses linear algebra. Also graphics and shadows in game use transpose matrices. A lot of algorithms use linear combination and other linear algebra.

Thanks guys for those comments, quite helpful.