Generating Sets

Definition

We say that the set ${v_1, v_2, \ldots, v_n}$ in the vector space $V$ is a generating set (or spanning set) for $V$ if:

That is, if any vector in $V$ can be written as a linear combination of $v_1, v_2, \ldots, v_n$.

Interpretation

A generating set for a vector space provides a way to "build" or "generate" every vector in that space using only linear combinations of the vectors in the generating set. In other words, the span of the generating set equals the entire vector space.

Examples

In $\mathbb{R}^2$

The standard basis vectors $e_1 = \begin{pmatrix} 1 \ 0 \end{pmatrix}$ and $e_2 = \begin{pmatrix} 0 \ 1 \end{pmatrix}$ form a generating set for $\mathbb{R}^2$.

Any vector $\begin{pmatrix} a \ b \end{pmatrix}$ in $\mathbb{R}^2$ can be written as:

In Polynomial Space

The set ${1, x, x^2, \ldots, x^n}$ forms a generating set for the vector space of polynomials of degree at most $n$.

Properties

1. Non-uniqueness: A vector space can have many different generating sets.

2. Minimal Generating Sets: A generating set that contains no redundant vectors (i.e., no vector in the set can be expressed as a linear combination of the others) is called a basis.

3. Finite Generation: A vector space is said to be finitely generated if it has a finite generating set.

4. Relation to Dimension: The minimum size of a generating set for a vector space equals its dimension.

Importance

Generating sets are fundamental in understanding vector spaces because:

1. They provide a way to describe all vectors in the space using a finite set of vectors.

2. They help determine whether a set of vectors spans the entire space.

3. They are used to find bases for vector spaces.

4. They connect to the concept of linear dependence and independence.

Exercise

Show that the set ${(1,0,0), (0,1,0), (0,0,1), (1,1,1)}$ is a generating set for $\mathbb{R}^3$. Is this a minimal generating set? Why or why not?