Computing projective coordinates and equation of a plane.

Question

The question

We are given a vector space with basis $\{e_1,e_2,e_3\}$ and a projective frame, $\{ [e_1],[2e_1 + e_2],[e_2 - e_3] \}$ and a fixed point $[3e_1 + e_3]$. What is the coordinates for the vector $[1,1,-1]$ ? What vector has coordinates $(2,0,1)$ ? Find the equation of the plane that is defined by the vectors given above ( Give both the parametric and cartesian equation).

My method ( or as outlined in our program )

I have found a matrix which takes the standard basis vectors $\{(1,0,0),(0,1,0),(0,0,1) \}$ to the corresponding vectors of the projective frame. Namely, the rows of this matrix are $( a_1,2a_2,0),(0,a_2,a_3),(0,0,-a_3)$. Then I have solved for the variables $a_1,a_2$ and $a_3$ which gave me the values $a_1 = 1,a_2 = 1,a_3=-1$. According to my notes, if I take the inverse of this matrix and take the product ( the dot product ) with the vector $[1,1,-1]$, I will get the coordinates.

Problems I have

1. For question 1,applying the above method gives me $(1,0,0)$,which I think is not correct. I have made a mistake and I cannot find it, any help would be appreciated.
2. For the second question, I multiplied the vector $(2,0,1)$ with the matrix that I found initially, that is before inverting it. Is this the right process ?
3. For the third question, I tried to use high school mathematics but it did not help. My problem is that donot understand the difference between parametric and cartesian equations. What makes one better then the other and how can I ( is it even possible ) to convert between them.

Background

I am a programmer ( not a mathematics graduate ) and the above question is from a 2-week training program, I am undergoing for graphics programming. The above question has been posed as "challenge" question to us ( I believe this is elementary for mathematicians ) and I would really like to understand how it works. I understand this is very elementary, but I have become confused with all the information available on the Internet and would really appreciate a straight answer.

Thank you all for your time and for sharing your knowledge.