Conflict of interests results in lies?

[seang]

Hello, I am currently studying Electrical Engineering. I have this class where we quickly go over selected math topics which will be oft used in our careers.

I'm also enrolled in Linear Algebra (not matrix algebra. The difference at my school is that Linear Algebra is based much more around proofs and theory, and matrix algebra is more application oriented; I'm not sure if this is the case most places.).

Anyway in my EE class, one of the topics was linear algebra. We basically learned everything we learned in my linear algebra class minus proofs and orthogonality.

So let me get to the point here. There was a true/false question on my EE exam that asked whether R2 was a subspace of R3. The answer accoring to my prof was true. However I remember my linear algebra teacher explaining to us why it wasn't. What's the story? Our linear algebra teacher said something about isomorphism, which we didn't cover in my linear algebra class, so I couldn't really argue much with my EE teacher.

Any clarification?

 

[matt grime]

Given a vector space isomorphic to R^3, then there are infinitely many non equal subspaces all isomorphic to R^2 - any choice of two linearly independent vectors will span one. However, there is in one sense no such thing as 'the vector space R^3', or 'the vector space R^2', though we often ignore this fact. The class of real vector spaces of dimension 3 is not even a set. So it is moot to ask 'is R^2 a subspace of R^3'. It is true, and no one can argue with this, that given a vector space of dimension n over a field F, then there are subspaces of dimension m for all 0<=m<=n.

 

[hypermorphism]

If by R^2, you mean a 2-dimensional vector space over R, and by R^3, you mean a 3-dimensional vector space over R, then the correct statement is that R^2 is isomorphic to a subspace of R^3 (one possible isomorphism is mapping all pairs (x, y) to 3-tuples (x, y, 0)), but R^2 is obviously not itself a subspace of R^3, since R^2 consists of 2-tuples and R^3 consists of 3-tuples.
There will be a lot more glossing over of isomorphisms when talking about physics/engineering.