To check if its your battery you could get a voltmeter and see the voltage. The battery alone should measure over 12V.

OR jump your car again and when its running unplug the (+) line and if its still running then its your battery. Not your alternator. Replace the battery, its not that expensive.

Im having this problem right now too. Waiting to get my Optima battery replaced. Nationwide outage of the yellow Top Optima due to quality control problems.