Definition of k-plane as linear subspace of dim. k?

Question

I am currently reading L.W. Tu's "Introduction to Manifolds" (2nd ed.). In exercise 7.8, which is concerned with showing that the Grassmannian $ G(k,\mathbb R^n) $ is a smooth manifold, Tu refers to the elements of the Grassmannian as k-planes through the origin in $ \mathbb R^n $.

My question is whether there is a rigorous definition of the notion of a "k-plane" of a vector space V with $ \text{dim} V \geq k$; for instance as an affine subspace of dimension $ k $? In other words, is it common to refer to $k$-dimensional linear subspaces of ANY vector space as a "$k$-plane" – in any context?

I would be grateful for some clarification! :)

Answer 1

I'd say in general it is an affine $k$-dimensional subspace of $V$. In this case $k$-planes through the origin are just the $k$-dimensional linear subspaces of $V$.