Let's delve into the fascinating world of matrix manipulation, specifically focusing on the concept of pivoting a matrix around a circled element. This operation is fundamental in various numerical ...

Most linear algebra courses start by considering how to solve a system of linear equations. \[ \begin{align} a_{0,0}x_0 + a_{0,1}x_0 + \cdots a_{0,n-1}x_0 & = b_0 ...

Some algorithms based upon a projection process onto the Krylov subspace $K_m = \operatorname{Span}(r_0, Ar_0, \ldots, A^{m - 1}r_0)$ are developed, generalizing the ...

Computer scientists have devised an innovative and elegantly concise algorithm that can efficiently solve systems of linear equations that are critical to such important computer applications as image ...

The class of second-order linear dynamical systems is considered. A method and algorithm are presented to transform any system with n degrees of freedom into n independent second-order equations. The ...

My understanding of linear independance is that if you have a unique solution after rref, it's linear independant. If you have multiple solutions or no solution, it's not linear ...