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Could be a number of things. First - check power steering fluid levels and look at what the oil looks like. If heavily oxidized and/or has a burnt odor, flush out with fresh fluid and bleed system for air.
If that checks all - verify that the serpentine belt is tracking on the power steering pump pulley. Might have to remove the serpentine belt, so that you can try and spin the pulley by hand. Note if there is any binding there or pulley side to side play.
Rubbing noise could be from binding in the steering column. Could be outside the passenger cabin or inside the passenger cabin. You'll have to look at it and see if you can pickup any excessive play or binding in the works.
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2002 Corolla S, 1.8L 1ZZ-FE VVT-i
2003 Matrix XRS, 1.8L 2ZZ-GE, VVTL-i (RIP)
2009 Matrix XRS, 2.4L 2AZ-FE VVT-i
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