Abstract Linear Algebra 4 | Basis, Linear Independence, Generating Sets

The core takeaway is that “basis,” “linear independence,” and “dimension” from standard linear algebra extend cleanly to abstract vector spaces—including infinite-dimensional ones—by defining everything in terms of finite linear combinations. That extension matters because it lets mathematicians describe polynomial spaces, function spaces, and other large objects using the same structural language as ${\mathbb{R}}^n$ and ${\mathbb{C}}^n$, even when the number of needed directions is unbounded.

The discussion starts by recalling the polynomial vector spaces $P_k$: all real-valued polynomials on $\mathbb{R}$ with degree at most $k$. As $k$ increases, these spaces grow, and in the limit there is no degree bound—leading to an infinite-dimensional polynomial space. That forces a broader notion of dimension than the usual finite one.

To set up the abstract framework, the transcript defines a general linear combination in a vector space $V$: given vectors $v_1,\dots ,v_K\in V$ and scalars ${\alpha }_1,\dots ,{\alpha }_K\in F$, a linear combination is ${\sum }_{j=1}^K{\alpha }_j{v}_j$. A key rule is emphasized: linear combinations are always finite sums. Even when the eventual vector space is infinite, the coefficients are chosen for only finitely many vectors at a time.

Next comes the span. For any subset $M\subseteq V$, $\mathrm{span}(M)$ is the set of all vectors obtainable as finite linear combinations of elements of $M$. The span is always a subspace of $V$. When $M$ is empty, $\mathrm{span}(\varnothing )$ is defined as the smallest subspace, containing only the zero vector.

A subset $M$ is then called a generating set for a subspace $U$ if $\mathrm{span}(M)=U$. This captures the idea that $M$ provides exactly enough information to build every vector in $U$ using linear combinations—no more and no less.

Linear independence is introduced as a uniqueness condition on coefficients. A set $M$ is linearly independent if the only way to represent the zero vector as a linear combination of elements of $M$ is the trivial one where all coefficients are zero. The set $M$ itself may be infinite, but the combinations tested are still finite.

Combining generating and independence yields the definition of a basis: a set $M$ is a basis of $U$ if it both generates $U$ and is linearly independent. The dimension $\dim (U)$ is the number of vectors in any basis of $U$; this number is fixed even though many different bases may exist. For infinite-dimensional cases, the transcript notes that one could use cardinality to distinguish different infinities, but for this course it will mainly distinguish only finite versus infinite, writing $\dim (U)=\infty$ when it is not finite.

Concrete examples anchor the definitions. For $P_0$, the basis is the constant function $1$, so $\dim (P_0)=1$. For $P_2$, the monomials $1,x,x^2$ form a basis, giving $\dim (P_2)=3$. The transcript also highlights that the space of all functions on $\mathbb{R}$ is infinite-dimensional. Finally, it points to a finite-dimensional matrix space: all complex-valued $2\times 3$ matrices have dimension $6$, and the task is to find a basis (a linearly independent generating set) with six elements, foreshadowing coordinate methods in the next video.