# Weird linear algebra question

So, I'm studying for my linear algebra midterm and I came up with kind of an interesting question that I pose to all of you brilliant people on physics forums.

Let's say you have a linear transformation T(x)=Ax, with A being an nxm matrice. Apparently, for this equation to hold, x must be a member of ℝm.

Maybe this is a ******** argument but if ℝm-1 is a subset/subspace (forgot the exact terminology) of ℝm then wouldn't the vector (2,1) in ℝ2 be (2,1,0) in ℝ3? vector operations with vectors in ℝm (or at least as far as I know) can't create vectors in ℝm+1, right?

I have no idea if I really made my question clear at all, but I'm curious to hear what you guys have to say regardless.

So, I'm studying for my linear algebra midterm and I came up with kind of an interesting question that I pose to all of you brilliant people on physics forums.

Let's say you have a linear transformation T(x)=Ax, with A being an nxm matrice. Apparently, for this equation to hold, x must be a member of ℝm.

Maybe this is a ******** argument but if ℝm-1 is a subset/subspace (forgot the exact terminology) of ℝm then wouldn't the vector (2,1) in ℝ2 be (2,1,0) in ℝ3?

Not necessarily. What's to say the vector in ℝ3 isn't (0, 2, 1) or (2, 0, 1)? In fact, it could be infinitely many different things, even using the same basis. Saying that ℝ2 is a subset of ℝ3 is analogous to saying that a given plane in a 3-D space is a subset of that 3-D space (granted it has to have some additional properties to be a subspace, like including zero). There are infinitely many different planes in ℝ3 that would qualify as a subspace, and any point in ℝ3 is inside a plane, so really (2,1) could represent infinitely many different points in ℝ3

vector operations with vectors in ℝm (or at least as far as I know) can't create vectors in ℝm+1, right? .

Yes they can! What if T(x) = Ax, where A is a matrix and x is a scalar? That goes from ℝ to a matrix space!

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