Question

How to determine if a vecter space in R3 is a subspace of a vector space in R4

Answer 1

A vector space in R3 will never be a subspace for R4. The first and foremost condition for being a subspace is that it should be a subset of R4. You should write a subset whose dimension is the same as R3 or one of the components is zero

If you include and zero vector and make your vector space a vector in R4 then you can use the following test to check whether its a subspace or not

If S is a subset of R4 then S is a subspace if

0 is in S

and for every x1,x2 in S and a c in field F . The element kx1+x2 is in S

Now getting back to your question you have stated a subset is a vector space in R3 so it must have zero element and second condition of subspace test is satisfied and hence it would be a subspace of R4 also ( if you add necessary zero elements to make it a subspace of R4)

If you need any further clarification regarding this problem feel free to comment below