0

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?

bfff
  • 139

1 Answers1

2

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.

Christopher A. Wong
  • 22,445
  • 3
  • 51
  • 82
  • So if we have a plane in $R^3$ we can still use a change of basis to map that plane into $R^2$? And am I correct to believe that this mapping is both onto and one to one? – bfff Oct 20 '20 at 06:45
  • 1
    Yes, that particular map between a plane in $\mathbb{R}^3$ and a plane in $\mathbb{R}^2$ is invertible. However, the problem is that this first plane is embedded in $\mathbb{R}^3$. When speaking about abstract linear maps, there is never such an embedding; the vector space in question is the entire space. But since matrices are this concrete representation of a linear map that works on coordinate representations, trying to treat the first plane as having 3 coordinates, when in truth it is a 2D vector space, creates a slight discrepancy that normally doesn't occur. – Christopher A. Wong Oct 20 '20 at 06:55
  • but then I get stuck because the matrix that corresponds to this mapping is not invertible. – bfff Oct 20 '20 at 07:44
  • A matrix only corresponds to the mapping when you restrict the image. Naively, a $3 \times 2$ matrix corresponds to a map $\mathbb{R}^2 \rightarrow \mathbb{R}^3$; however, you want to represent a map $\mathbb{R}^2 \rightarrow W$ where $W$ is a 2D subspace. Suppose we have a linear map taking a vector representing 3D spatial coordinates and maps it to some other 3D space. So you get a $3 \times 3$ square matrix. Somebody objects "Wait! Our world is actually 11-dimensional due to string theory, so the matrix should be $3 \times 11$!" Clearly this should not affect our $3 \times 3$ construction. – Christopher A. Wong Oct 21 '20 at 08:18