Definitions (summarized from 3blue1brown)
- span - the set of all possible vectors you can reach with a given set of vectors
- linearly dependent vector - a vector that can be expressed as a linear combination of the others
- basis vectors - a set of linearly independent vectors that span the full vector space
- determinant - Think of the columns of the matrix as vectors at the origin forming the edges of a skewed box. The determinant gives the volume of that box. reference
- inverse of a matrix - if you multiply a matrix by this, you will undo its transformation and get the identity matrix (if the determinant is zero, then there is no inverse, because you can't unsquish a 2D line and get a 3D space)
- rank - number of dimensions in the output of a transformation (if output of transformation is a line it's rank 1, plane -> rank 2, etc)
- full rank - when rank == number of columns (rank is as high as it can be)
- column space - the set of all possible outputs of the matrix
- null space - the space of all vectors that become the zero vector after you apply the matrix
- eigenvector of a matrix - this is a vector that stays within its own span when we apply the matrix
- eigenvalue of a matrix - the factor by which the eigenvector gets stretched after the transformation
- diagonal matrix - all elements of matrix are zero except diagonal. all the columns are eigenvectors for such a matrix