Show: If U is a subspace of R^n with dimension k<n, then U is an empty set.
Not true! Consider the cartesian plane $\displaystyle \mathbb{R}^2$, any line through origin is a subspace and has dimension 1 (which is less than 2).Its a non empty set.
Not true! Consider the cartesian plane $\displaystyle \mathbb{R}^2$, any line through origin is a subspace and has dimension 1 (which is less than 2).Its a non empty set.
Originally Posted by TXGirl
Show: If U is a subspace of R^n with dimension k<n, then U is a null set (measure theory).
She meant "null measurable set" not "empty set". This is still not true. I gave a counterexample to that already. Just consider a line segment. A line segment does not have null measure.