Yes... kind of. You'll be projecting the image to a larger size.

If you move the lens out by a 1/3 of focus distance using extensions, shouldn't you gain the full frame image? With possible loss of infinity focus?

Putting 35mm of extension on a 100mm lens or 10mm of extension on a 30mm lens isn't just a "you can't focus at infinity" but takes you well into the range of "you can't focus on anything from one foot to infinity" range.

You will also lose some light. fe = fa * (1 + m) gets us to effective aperture is equal to the actual aperture (lets say f/2.8) * one plus the magnification ratio. The magnification ratio, as described will be 1/3. This gives us fe = 2.8 * 4/3 = 3.7. So, you're going to lose about a stop of light in this too. If your lens is an f/5.6 rather than an f/2.8, many autofocus systems will fail to work at when additional light is lost.

This may be an acceptableish solution for if you wanted to do macrophotography without buying new glass, but this setup won't be useful for much of anything else.

Furthermore, it should be pointed out that the edge of the image is where the most aberrations are. Using extension tubes or similar on a full frame lens on a full frame system helps reduce the aberrations in the glass by utilizing the best part of the lens. Its not perfect, but it helps.

Using an APS-C lens for macro would enlarge the aberrations and still have the edge of the image circle in the frame. This may result in unacceptable edge image quality.