This video should help in only 90 seconds!
Check out this video I made called "Swimming in an Ocean of Air" which explains why air is matter.
You will see a demo where I have evacuated the air from a vacuum chamber. The chamber is on a scale, and when I let the air back in, you can see how much the air in the chamber weighs.
By the end, I should convince you that air is matter. It is a physical substance that takes up space and indeed, has mass.
And on top of all that, using physics, I answer the philosophical question:
Is the glass half empty or half full?