While trying to understand local chart on Grassmannians I came across this example in this book:
Take $V = \mathbb R^2$ and $U,W$ two subspaces generated by linearly independent vectors. The books gives the following picture:
But they write:
"The point $P$ in the sketch belongs to the subspace $W_A$. It is specified uniquely $u + Au $ with $A$ a real number ($1 \times 1$ matrix) and $u \in U$. ..."
But I don't see how it's possible for $Au$ to lie outside $U$ and clearly, $W_A$ and therefore $P$ lie outside $U$. Please could someone enlighten me what is meant here? How can $Au$ be in $W$ when $A \in \mathbb R$?
I think things are not phrased too nicely in the book you refer too, since it uses a mix of the "abstract" approach to linear algebra and the approach via matrices.
If you prefer the abstract approach, then you should say that given a subspace $W\subset V$, you find a local chart for the Grassmannian around $W$ by picking a subspace $U$, which is complementary to $W$ in $V$. Then the domain of this chart consists of those subspaces of $V$ of the same dimension as $W$, which intersect $U$ in $\{0\}$. The chart itself is a bijection between this neighborhood and the space $L(U,W)$ of linear maps. This bijection is simply given by viewing the graph of a linear map $f:U\to W$ as a subspace in $U\times W=V$. Conversely any subspace intersecting $U$ in $0$ only can be interpreted as a graph.
This description is basically the one used in the book in the case that $V$ is $\mathbb R^2$ and $W$ is one-dimensional. It just gets confusing, when the author says that now you view the linear map $A$ as a $1\times 1$-matrix and hence as a number. On the one hand, this requires a choice of bases in the two subspaces, which is not made. On the other hand, $A$ is a linear map between two different spaces of dimension one, so it is NOT multiplication by a real number. So the confusion mainly is between a linear map, its graph and its representation as a matrix.
To write out the matrix version nicely I would just say that in $\mathbb R^2$ you can define a chart on the subspace of lines which intersect the $x$-axis in $0$ only (i.e. which are not horizontal) by mapping to the unique value $y\in\mathbb R$ such that the line contains the point $(1,y)$. This exactly corresponds to the above description using $(1,0)$ and $(1,0)$ as the bases of the coordinate axes, which are the complementary lines chosen initially.