Hi, I have some questions about basis.

1-)Let's say w_{1},w_{2},......,w_{L}is in the Span(U), (U={u_{1},u_{2},.....,u_{K}} , u_{1},u_{2},.....,u_{K }are the basis of U)

If L>K then w_{1},w_{2},......,w_{L }is linearly independent.

I understand the statement intuitively very good, but can someone give me a formal proof? I don't feel fully convinced.

2-) If we have a Vector Space U, and its basis consists of , lets say n vectors, all other bases that I will find for the vector space U will consist of n vectors? I can't find more linearly independent vectors that spreads the whole vector space that is also the base?

Thanks you