Projective space and its basis I am trying to solve an exercise from the book "Permutation Groups" by J. Dixon and B. Mortimer. 

Later, I asked a similar question about the basis of Affine geometry " Affine geometry and its basis". Similar to the answer of that question, I think that in this case, all of the basis of $PG_{d-1}(F)$ are of the form  $\{[v],[v+v_1],[v+v_2],\ldots,[v+v_d]\}$, but I cannot prove that why the other forms cannot be a basis. Also, I have a problem with showing that $PGL_d(F)$ is transitive over the basis of the above forms. Another form that I think can be the form of all such basis is $\{[v_1+v_2+\cdots+v_d],[v_1],[v_2],\ldots,[v_d]\}$, but I don't know it is true or not.
I appreciate your help.

I don't think either of the two forms of a projective basis that you suggest is quite right.
By definition, such a basis has the form $\{[v_1],\ldots,[v_{d+1}] \}$, subject to the condition that any $d$-element subset of $\{v_1,\ldots,v_{d+1} \}$ is linearly independent (and hence is a basis of $F^d$). This condition  is equivalent to $\{v_1,\ldots,v_d \}$ being linearly independent, and $v_{d+1} = \lambda_1v_1 + \cdots + \lambda_d v_d$ for some $0 \ne \lambda_i \in F$.
Given two such projective bases $\{[v_1],\ldots,[v_{d+1}] \}$ and $\{[v'_1],\ldots,[v'_{d+1}] \}$, with $v_{d+1} = \lambda_1v_1 + \cdots + \lambda_d v_d$ and $v'_{d+1} = \lambda'_1v'_1 + \cdots + \lambda'_d v'_d$, you can check that the elements of ${\rm GL}_d(F)$ that map the first to the second are of the form $v_i \mapsto \alpha \lambda'_iv'_i/\lambda_i$ for some fixed nonzero element $\alpha$ of $F$. So there is a unique element of ${\rm PGL}_d(F)$ which does this.
