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.
NO. Independent vectors means if then . They are independent because if then then thus .
I gave you an example. Consider it is impossible to express because not matter what you will never be able to get a non-zero entry in the first coordinate. So it cannot span R^3.but why do those two vectors not span R^3...I don't get it