Tutor profile: Maryel S.

What is a basis?

A basis of a vector space $$V$$ is a subset $$B\subseteq V$$ that is both a spanning set and a linearly independent set. In other words, a subset $$B \subseteq V$$ is a basis of the vector space $$V$$ if $$B$$ is a linearly independent spanning set of $$V$$. This means that any element of $$V$$ can be expressed as a linear combination of the elements of $$B$$, and the only way to express $$0$$ as a linear combination of the elements of $$B$$ is by assigning each coefficient in the linear combination to be $$0$$.

What is De Morgan's Law?

De Morgan's Law basically says that for any two sets $$A$$ and $$B$$, the following equalities hold: $(\left(A \cap B\right)^c = A^c \cup B^c \text{, and } \left(A \cup B\right)^c = A^c \cap B^c.$) De Morgan's Law also has many applications in fields other than Set Theory (Probability, Statistics, etc.)

What does the following statement mean? $(\lim_{x \to a} f\left(x\right) = L$)

This means that for every $$ \epsilon >0$$, there exists a $$\delta>0$$ such that $$\left|f\left(x\right)-L\right|<\epsilon$$, whenever $$\left|x-a\right|<\delta$$. This just means that when the value of $$x$$ is close to $$a$$, the value of $$f\left(x\right)$$ is close to $$L$$.