How to make sure that my lecture notes do not infringe on copyrights?
In the case of using problems from a text book, you would not need to worry about copyrights as long as you cite it (whether you change the numbers or not). It would be good to include a references section with the bibliography information of the book included in it.

Edit:

Academic textbooks are generally meant to be reusable for educational purposes. Having stated the above, depending on the country and the copyrights explicitly mentioned in the textbook, you might have to obtain prior permission from the publisher or author. (Please refer the discussion in the comments for details)