As you know, the definition of a basis is that it is an independent spanning set,

which means you need two condition for it to be true...

if U is a subspace of R^n, and B={x1,x2,...,xn}

1.B must be linearly independent

2. B is a spanning set of U

But the thing I don't understand is that how could you have a linear independence for B without having B being a spanning set of U.

for me this is kind of confusing...mostly since I can't really visualize these types of algebra concepts like i do with calculus.