I am going through 3b1b linear algebra video, and at the end of 2nd video in the playlist, he defines basis as The basis of vector space is a set of linearly independent vectors that span the full space.
My question is, isn't it redundant to say the part that span the full space because a set of linearly independent vector always span a subspace or a space. For example, if we have two 3×1 independent vectors then their span a 2d subspace in a 3d vector space.
So, why it is not sufficient just to define a basis as a set of linearly independent vectors?
Best Answer
Let $V=\mathbb{R}^3$, the vectors $(1,0,0)$ and $(0,1,0)$ are linearly independent. As you say they certainly form a basis for some space, namely $\mathbb{R}^2\times\{0\}$. They do not however form a basis for all of $\mathbb{R}^3$.