The many uses of singular vectors

Blah blah

blah blah blah

Important notes

This is the text on the left side of the margin.

Questions

This is the second thing I wish to write. It will be a boring paragraph.

The third thing I wish to write is this. Which came before the previous thing.

The determinant

A matrix can be used to change from one co-ordinate system to another. However, we would like to know how shapes in co-ordinate system A will be changed when transformed to co-ordinate system B. If we focus on one shape in particular, the unit n-cube, then we can see how the volume changes after we apply our linear transformation by examining the determinant of the matrix in question.

The determinant can be calculated by taking the product of each of the eigenvalues of our matrix. The volume of a unit $n$-cube is calculated by taking the length of its side to the power of $n$. The eigenvalues also tell us how much the corresponding eigenvector is scaled after applying our matrix. We can triangularize our matrix such that it’s diagonal elements consist of its eigenvalues. The determinant of a triangular matrix is the product of the eigenvalues. Now if the determinant is the product of the eigenvalues, then the eigenvalues tell us something about how shapes change in the new space.

Yap