I’ve been working with two definitions of lines in $\mathbb{P}_\mathbb{R}^2$, and tried to show their equivalence.
The first is that, given two points $a=(a_0:a_1:a_2)$ and $b=(b_0:b_1:b_2)$, the line between them is given by $\{ua+vb:u,v\in\mathbb{R}\}$.
The second is $$\{(X_0:X_1:X_2):k_0X_0+k_1X_1+k_2X_2=0\}$$ for some $k_i\in\mathbb{R}$ not all $0$.
Given two distinct points $a$ and $b$, we can use simple linear algebra to find $k_i$ such that our points lie on that line.
However I've been struggling to show the converse. That is, given two distinct points $a=(a_0:a_1:a_2)$ and $b=(b_0:b_1:b_2)$ such that $$k_0a_0+k_1a_1+k_2a_2=0$$ and $$k_0b_0+k_1b_1+k_2b_2=0$$ for some $k_i\in\mathbb{R}$ not all $0$, then for any point $c=(c_0:c_1:c_2)$ such that $$k_0c_0+k_1c_1+k_2c_2=0$$ we should be able to write $c=ua+vb$ for some $u,v\in\mathbb{R}$.
Perhaps I'm overlooking something simple, but any help would be much appreciated.
Since $(k_0,k_1,k_2$) isn't in its own orthogonal complement it can't be of dimension $3$. Then because both $a$ and $b$ are, and are linearly independent, it must be of dimension $2$ with $a$ and $b$ as a basis. Then since $c$ is in the orthogonal complement it must be expressible in terms of $a$ and $b$, and we can use simple linear algebra to find the coefficients.
Thanks to amd for their comment.