Least squares and pseudo-inverses

To appreciate the connections between solutions of the system \(Ax=b\) and least squares, we begin with two illustrative examples. Overdetermined systems: \(A\) is \(m \times n\) with \(m > n\). In this case there are more equations than unknowns, \(A^{\top}A\) is \(n\times n\) and \(AA^{\top}\) is \(m\times m\). The connection with the pseudo-inverse is that … Read moreLeast squares and pseudo-inverses

Intermediate Value Theorem – Limits and Continuity

Intermediate Value Theorem To begin with, let’s start with the basic statement of the theorem. Theorem If \(f(x)\) is continuous on a closed interval \([a,b]\) and \(N\) is any number \(f(a) < N < f(b)\) then there exists a value \(c \in (a,b)\) such \(f(c) = N\). The illustration corresponding to the theorem is to … Read moreIntermediate Value Theorem – Limits and Continuity