Notation from book:
$F$ means $\mathbb{R}$ or $\mathbb{C}$. You should think it means the real numbers when you read it. I will write all of these notes using $F=\mathbb{R}$, but you could use $\mathbb{C}$ just the same.
$V$ is always a vector space over $F$.
Linear combination of $m$ vectors $v_1, \ldots, v_m$ is a vector of the form
$$ a_1 v_1 + \cdots + a_m v_m, $$ where all of the $a_i$ are real numbers.
Span. The set of all linear combinations of a list of vectors is called their span. The span of the empty set is defined to be ${0}$. In notation,
$$ \Span(v_1,\ldots,v_m) = \left\lbrace a_1 v_1 + \cdots + a_m v_m \right\rbrace , $$
where the $a_i$ are real numbers.
A set of vectors “spans” a vector space if their span is the whole vector space.
All lists of vectors in this book are finite length.
A finite dimensional vector space is spanned by a (finite) list of vectors.
A vector space is “infinite dimensional” if it is not finite dimensional.
Polynomials of degree at most m are denoted $\mathcal{P}_m$.
polynomials of any degree is $\mathcal{P}$.
Linear independence. A list of vectors $v_1, \ldots, v_m$ is linearly independent if the only choice of $a_1, \ldots, a_m$ that makes $$ a_1 v_1 + \cdots + a_m v_m = 0$$ is $a_1=\cdots=a_m = 0$.
Linearly dependent. You can find some $(a_1,\ldots,a_m)$ not all zero that makes the “linear dependence equation” be true: $$ a_1 v_1 + \cdots + a_m v_m = 0$$
The span of a list of vectors in $V$ is the smallest subspace of $V$ containing all of the vectors.
Linear dependence lemma (to prove).
Length of lienarly independent list <= length of spanning list.