• Reflectors are a class of matrices that are not introduced in all linear algebra textbooks. However, the book of Carl D. Meyer uses this class of matrices heavily for various fundamental results. Indeed, the book uses reflectors for theoretical reasons, such as proving the existence of fundamental transformations like SVD, QR, Triangulation, or Hessenberg decompositions (more to come below), as well as applications, such as the implementation of the QR algorithm via the Householder transformation or solution of large-scale linear systems…

• The four fundamental spaces of a matrix $A$, namely the range and the nullspace of itself $A$ or its transpose $A^T$, are the heart of linear algebra. We often find ourselves in need of computing a basis for the range or the nullspace of a matrix, for theoretical or applicational purposes. There are many ways of computing a basis for the range or nullspace for $A$ or $A^T$. Some are better for application, either due to their robustness against floating point…

• There are at least two answers to this question; one of these is more educative and the other one is at least as educative (in a different and profound way) as well as practical. Method 1 The first method is a more introductory level method. It is helpful to know it and good to read it as a refresher even if one is more advanced student of the topic. A linear system $\mathbf{Ax=b}$ is homogeneous when $\mathbf b=\mathbf 0$ and nonhomogeneous…