Looks like you already have a rack on your bike, to carry that trunk bag thing...so your best bet is to get a set of panniers to hang on it. (Or just one pannier, for that matter.) That will give you more than enough space for a change of clothes, and lots of other stuff besides.

The more you commute, the more you'll be glad for the extra space, both for times when you need to carry extra stuff for work, and when you want to do errands or shopping before you go home.