The change of basis matrix(skip to 9:00) used in this video is 3x2 and clearly not invertible: https://www.khanacademy.org/math/linear-algebra/alternate-bases/change-of-basis/v/linear-algebra-change-of-basis-matrix
yet, the textbook I am using and posts like this one: Rigorously proving that a change-of-basis matrix is always invertible
state that the change of basis matrix is invertible.
If we have a plane that is a subspace of $R^3$, clearly the change of basis matrix is going to be 3x2.. Again, a 3x2 matrix is clearly not invertible.
Can somone please explain?
Normally, a change of basis matrix refers to a matrix that maps a basis in $\mathbb{R}^n$ to another basis in $\mathbb{R}^n$. In this case, then the change of basis matrix is square and invertible.
However, it seems that in that video, they are transforming between two bases in a subspace of $\mathbb{R}^3$ rather than the whole space. In this case, then the matrix will not be invertible; however I would say that this is not the normal usual way of defining the change of basis matrix.