I am currently majoring in Physics but I am thinking about changing my major to Mathematics because feel more attracted to it as a subject. However, I have been told that in Physics, one almost always gets a full scholarship to attend graduate school. Is it hard to obtain a scholarship to go to grad school in Mathematics here in the U.S.?
At most US universities, Ph.D. students in mathematics usually receive full financial support that covers tuition as well as a modest stipend (i.e. money paid to you directly for your living expenses, usually between $15,000-25,000 per year). This is normally some combination of:
Teaching assistantship: requires that you teach. Often this starts with discussion sections for calculus and other introductory courses, and can progress to more specialized courses.
Research assistantship: requires that working on research (usually your own thesis research, but sometimes a different project). This often comes from your advisor's grant funds.
Fellowship: doesn't require you to do any extra work, just your usual coursework or research.
In my experience, the word "scholarship" is not usually used when talking about funding for graduate study.
Note that the above only applies to Ph.D. programs. Masters students generally have to pay their own tuition and living expenses (perhaps via loans), though sometimes a limited number of teaching assistantships, etc, are available.