I am reading "Algebraic Geometry: A First Course" by Joe Harris and frankly I am very confused about projective $k$-planes, or maybe in general how points in projective space behave. I understand from the definition that $\mathbb{P}^n$ is simply defined as $K^{n+1}/\sim$ where $x\sim \lambda x$ for all $\lambda$, so is it normal to visualise a point in $\mathbb{P}^n$ as a line in $K^{n+1}$ passing through the origin?
Similarly, as the book proceeds to defining $k$-planes, it says
An inclusion of vector spaces $W\cong K^{k+1}\to V\cong K^{n+1}$ induces a map $\mathbb{P}W\to \mathbb{P}V$; the image $\Lambda$ of such a map is called a linear subspace of dimension $k$, or $k$-plane, in $\mathbb{P}V$. In case $k=n-1$, we call $\Lambda$ a hyperplane. In case $k=1$ we all $\Lambda$ a line.
Intuitively, as points in $\mathbb{P}^n$ are lines through origin in $K^{n+1}$, is it safe to imagine a line in $\mathbb{P}^n$ as a two-dimensional plane through origin in $K^{n+1}$? And in general how do we handle with hyperplanes? For instance in Theorem 1.4 of the book, one of the sentence states
if $\Gamma=\Gamma_1\cup\Gamma_2$ is any decomposition of $\Gamma$ (a collection of $2n$ points in general position) into sets of cardinality $n$, then each $\Gamma_i$ will span a hyperplane $\Lambda_i\subset \mathbb{P}^n$.
Why is the last sentence true?
Sorry if this question seems a little bit vague. Thanks for any help.