## 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

## Continuity – Limits and Continuity

Continuity A function $$f(x)$$ is said to be continuous at a point $$a$$ in its domain if the following three properties hold. $$\displaystyle \lim_{x \to a} f(x)$$ exists. This takes three steps to show in itself. $$f(a)$$ has to exist, $$\displaystyle \lim_{x \to a} f(x) = f(a)$$. Continuity connects the behaviour of a function in … Read moreContinuity – Limits and Continuity

## Strategy to Calculate Limits – Limits and Continuity

Strategy to Calculate Limits   To compute $$\displaystyle \lim_{x \to a} f(x)$$: A. Try to plug the value of $$a$$ directly into the function. If we get a number or the limit ‘blows up’ then we are done! You should be so lucky. Typically the value is undefined, having the form $$\displaystyle \frac{0}{0}$$ or \(\displaystyle … Read moreStrategy to Calculate Limits – Limits and Continuity