In $\mathbb{R^3}$, when we are given the span of $2$ vectors, we can describe the Span geometrically by
-a line if one of these vectors is a multiple of the other (since a line is the Span of all scalar multiples of a single vector.)
-a plane if they are not scalar multiples of each other.
Q) Now if we have $3$ vectors, could we describe their span geometrically (if the set of vectors making up this span are linearly independent) as a sphere, that is with infinite volume?
If we go to $4$ linear independent vectors in $\mathbb{R_4}$, would their spanning set occupy the entirety of the four dimensions, and so on? It makes sense to me, but I might be making an error, and I don't want to get the wrong ideas.
Please correct me if this question is not clear enough, nevertheless, thank you.
- I edited the question as I made a stupid mistake, which I noticed from the answer, I failed to move on to $\mathbb{R_4}$ when extending to $4$ vectors, that wasn't the meaning of my question at all so I just edited it.
Best Answer
As Tanner commented, The span of three linearly independent vectors in $\mathbb{R^3}$ is the entirety of the vector space $\mathbb{R^3}$. Geometrically, this means that the vectors span all of three-dimensional Euclidian space. A sphere of infinite volume is technically the same, but I think it's best to think of it just as the entire space.
If the vectors are not linearly independent, then the span will just be a plane or a line. There are no sets of four linearly independent vectors in $\mathbb{R^3}$, but in $\mathbb{R^4}$, four linearly independent vectors would span the entirety of that (four-dimensional) space, as you guessed.