Prove for any line $\{ a +tv \mid t \in k\}$ the embedding $\{[1: a_1+ tv_1: a_2+tv_2: ... : a_n +tv_n] \mid t \in K\}$ is a line.

Question

I am working through algebraic geometry and trying to understand the field by working through questions of my own. If I define $\mathbb{P}^k \subset \mathbb{P}^n $ to be the projective image of a $k+1$ dimenisonal subsapce of $K^n$. Then are we able to prove any line $\{ a +tv \mid t \in k\}$ the embedding $\{[1: a_1+ tv_1: a_2+tv_2: ... : a_n +tv_n] \mid t \in K\}$ is a 1 dimensional projective space.

Also,

1. If true, the does the projection of the vector space k_1(1,a_1,a_2,....,a_n) + k_2(0,v_1,...,v_n) contain our set.
2. Are they the same. (Not clear in my drawings) Can you prove this?

Look forward to much of your help. Just trying to gain a deeper insight into projective geometry because I do struggle with their definition of subspaces.

Answer 1

Yes, as any two dimension vector subspace can be decribed by $$ k_1a+k_2b$$ for $k_1,k_2 \in K$. Thus, any line projective line is $$\{ [a_0+k_2b_0: a_1+kb_2:...:a_n+bk_n] \} $$, as required.