Question

Use your growing understanding of subspace... Can R⁴ be a subspace of R³? Explain.

Answer 1

No. Because for R^4 to be a subspace of R^3, first of all the former should be a subset of the later directly or somehow embedded which is not the case. There are other liner algebraic explanation like the dimension of a subspace of a vector space can be at most the dimension of the super space. But here dim(R^4)=4>dim(R^3) which is a contradiction. Dimension of a vector space means the number of basis elements in a basis.

If you still have any doubt you can tell me in the comment.