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I was just skimming through some pages on the Gram-Schmidt process, and because I hadn't revised my linear algebra in a while I actually forgot (and I still don't know) some basic stuff:

They used in an example 3 vectors with each of them 4 entries, and they said that V is the subspace spanned by those 3 vectors. I have a couple of questions regarding this:

  • Is this in $\mathbb{R}^4$ or in $\mathbb{R}^3$?

  • What kind of object is created by these vectors? 1 vectors can create a line (a vector with its scalar multiples), 2 vectors create a plane, but does 3 vectors just create a cuboid?

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Vectors with $4$ entries live in $\Bbb{R}^4$, so $V$ is a subspace of $\Bbb{R}^4,$ which most often will be 3-dimensional. Specifically, if the $3$ vectors are all linearly independent, then they span a linear subspace isomorphic to $\Bbb{R}^3$. I don't know a geometric term analogous to "line" or "plane" for these 3-D objects; usually one just calls $k-1$-dimensional subspaces of a $k$-dimensional vector space hyperplanes, regardless of $k$.

It's certainly not a cuboid as I would use the term, since such a polyhedron is bounded.

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