After going through the process of learning all the different varieties of chords and arpeggios in root position on your instrument, the next step is to learn all of the inversions of each chord.
I was taught to learn the arpeggio inversions in four variations. Always start on the index finger and stop at the highest note you can reach in each position. Inversions are permutations of chord tones. This method of study helps you learn the information all over the neck of the bass. Learn to see the four shapes that are formed and be sure to repeat the same inversions high on the neck.
The examples below all start in the key of F. Remember to start with the index finger for each pattern. Each inversion starts on the E string and ends on the high G string. After playing the arpeggio in root position, shift the index finger to an “A note” (5th fret on the E string) and end each pattern on the pinky of the left hand. Then shift the index to the “c note” (8th fret on the E string) for the second inversion. Shift the index to the high “e note” (12th fret on the E string) for the third inversion.
Explore other chords and inversions such as the augmented 7, minor/major 7, minor 6, and augmented-major 7. Apply the same drill to each new chord you learn. Play the same chords and inversions on the piano. All of these examples are in the key of F. Work on all the chords in all keys. Break the information down to one key a week or something that works for you. This is a true, no-nonsense study of music theory on your instrument. The time that you put in will be time well spent!