Dimension of U,V and W over K: Do they Equal?

[garyljc]

Just have a question
if U,V and W are over the same field K
does it mean that dim U = dim V = dim W ?

 

[Ben Niehoff]

No.

For example, $R^2$ and $R^3$ are both over the same field R, but R^2 has dimension 2 and R^3 has dimension 3.