¿Cuál es la diferencia entre “to improve knowledge” y “to increase knowledge” ?Puedes dar oraciones como ejemplo.

There isn't much of a difference in these two, as both sound just as natural in any sentence. The difference between "improve" and "increase" is that improve means to better something, such as "I improved at doing my math homework.", while increase means to raise, as in a quantity. Because knowledge is a skill, it can be bettered, however it can also be considered a quantity, like your intelligence stat in a video game, and can be increased. You would not, however, say "I increased at my math homework." as it implies that a quantity is increasing, and it's uncertain what you increased. If you said "I increased at my skill at doing math homework", you give a quantity that you increased (your skill at math homework). In the "I improved at my math homework", you still need something to improve at, and the words "improve at" imply your skill increasing. Another example would be "I improved the dam", which would mean that the dam is better in some way (to which one might ask "in what way was it improved?") while "I increased the dam" would mean that the quantity of the dam increased, maybe meaning you made it bigger. In this case "I increased the strength of the dam" would mean nearly the same thing as "I improved the dam"