How do I determine if a subset is a vector space/linear space living in the ambient space of a R3? How do I guess the dimension?

Question

I'm not really sure what the first step is. I know that we have to follow the three rules.

1. Having a 0 vector in the space
2. Closed under scalar addition
3. Closed under multiplication

The problem I'm confused with {(x, y, z) ∈ R3 | y = x + 1}

Please don't just give me the answer or do the problem for me. I'm just looking for what the first step is.

I know we have to get into a matrix, but I'm just a little confused on how. The matrix represents the vector space right? I'm also confused about what Closed under scalar addition and Closed under multiplication mean.