TITLE: What is the dimension of this Grassmannian?
QUESTION [13 upvotes]: Why is  $2\times 3$ the dimension of $Gr_2(\mathbb{R}^5)$? and can one use the dimensions of Lie groups to derive this dimension?
Note: $Gr_2(\mathbb{R}^5)$ denotes the Grassmannian of all $2$-dimensional subspaces of $\mathbb{R}^5$.

REPLY [16 votes]: The easiest proof is this: to give a $k$-plane in $\mathbb R^n$ you must give a $k \times n$-matrix $M$, hence $kn$ variables. But this is only unique up to multiplication by invertible $k \times k$-matrices, so you must subtract $k^2$. Hence the dimension is $kn-k^2=k(n-k)$. In your case, it is $2 \cdot (5-2)=2 \cdot 3$.