Suppose that linear space L has a basis with a finite number of points, B = {P1,
P2, P3, … Pn} Then every linearly independent set is contained in a
basis.
I'm not sure how to go about proving this. I know that no linearly independent set can be larger than basis B, but I'm not sure how to prove there isn't a linearly independent set outside of the basis.
Best Answer
You show this by simply adding vectors to some linearly independent set $X$ until you find a basis. You start with some linearly independent set $X = X_0$, and $k = 0$.
Since $L$ is finite-dimensional, this algorithm must stop at some point, because if $L$ has dimension $n$, no set with more than $n$ elements can be linearly independent. But if the algorithm stops during iteration $k$, then $X_k$ is a basis, and by construction $X_k \supset X$.
Therefore, whenever we're given a linearly independent set $X$, we can run this algorithm to find a basis $X_k$ which includes $X$.