Linear Algebra

There is another, more insightful way of looking at a matrix. It turns out that that perspective, is the foundation of most of the complicated matrix manipulations.

Through different ways of matrix multiplication, we see the nature of matrices and how they work. This provides a crucial perspective on how matrices operate as linear objects.

When does Ax=b have a solution? Column-Space and Null-Space provide a new perspective for not only solving a system of equations but also to view a matrix.

Every matrix is a linear traforamtion that maps the input space into outputspace and devides each one into two complement subspaces. By knowing the rank of a matrix, one can determine the structure of its fundamental subspaces and predict the behavior of Ax=b.

with the help of fundamental subspaces we decompose vector b into two orthogonal components in column space and left null space, to solve Ax=b for those vector b that are not in column space of A. 

Although multiplying by Aᵀ solves the least‑squares problem algebraically when b is not in the column space of A, it is numerically unstable. QR decomposition not only provides a far better alternative, but also opens the door to more advanced methods

When matrices are square, everything feels more tangible. Most of our systems of equations lead to square matrices of coefficients. Eigenvalues are single numbers that reveal the nautue of a whole matrix. 

By diagonalization, we rewrite the square matrix in its best possible form.

By introducing SVD, we obtain the most general and powerful approach for solving Ax = b.