How could a 3 by 2 matrix have a unique solution in R3 for (Ax = b)?

[Science Student]

The actual question implies that this is possible, but I am just curious how.

 

The rows in R2 are only in two dimensions. So geographically, this seems to mean that the point where the lines cross would still be in R2, right?

 

Or am I misunderstanding what it means to "be in R3"?

[John]

If I'm understanding you correctly, then the short answer is "yes," the point of intersection will be "the same."

The longer answer is that while R² is a subset of R³, vectors in R² aren't really the same objects as vectors of the form <x, y, 0> in R³. However, R² is isomorphic to (i.e. has the same structure as) the subspace of R³ containing these vectors, so results in one will "carry over" to the other.

I hope that explanation isn't too muddy.

Edited May 7, 2014 by John

[Science Student]

If I'm understanding you correctly, then the short answer is "yes," the point of intersection will be "the same."

 

The longer answer is that while R² is a subset of R³, vectors in R² aren't really the same objects as vectors of the form <x, y, 0> in R³. However, R² is isomorphic to (i.e. has the same structure as) the subspace of R³ containing these vectors, so results in one will "carry over" to the other.

 

I hope that explanation isn't too muddy.

That makes sense; thanks a lot!