I have done some similar questions, for example the following:
Verify if three 3-dimensional vectors (v1:[…],v2:[…],v3:[…]) will span $\mathbb{R}^3$.
All I have to do is to put three vectors into a 3 by 3 matrix and perform elementary row operations, and to check if there are 3 pivots.
However, this makes me think if I have two 3-dimensional vectors and it does have a reduced row-echelon form with 2 pivots. Can I say that they span $\mathbb{R}^2$
More generally, does the dimension of vectors limit the space that the set of these vectors can span? (Ex: Can three 4-dimensional vectors span $\mathbb{R}^3$)
Best Answer
The first sentence after the colon is incomplete. However, if I get what you're driving at, the answer is yes that $k$ vectors in $\Bbb R^n$ will, if they are independent, span a $k$-dimensional subspace.
Recall that the number of vectors in a basis is an invariant; and that this number is called the dimension.
Moreover, any two $k$-dimensional vector spaces are isomorphic. So if we have $k$ linearly independent vectors, they span a space isomorphic to $\Bbb R^k$.