Previously, we defined a vector space broadly as a set which is closed under addition and scalar multiplication. This definition works to identify \(\mathbb{R}^n\) as a vector space, however, vector spaces are actually more general than just for collections of real numbers.
The vector space \(\mathbb{R}^n\text{,}\) and its operations of addition and scalar multiplication satisfy certain basic and simple properties. In fact, many other mathematical objects, such as matrices, polynomials, and functions, also satisfy these properties, given suitable definitions of addition and scalar multiplication. This leads to the more abstract definition of a vector space, that encapsulates all of these ideas.
Subsection 8.1.3 Subspace Spanned by a Set
First, we define analogous notions for an arbitrary vector space. Let \(V\) be a vector space.
Definition 8.1.5.
Let \(\vec{v}_1, \dots, \vec{v}_n \in V\) be vectors, \(a_1, \dots, a_p \in \mathbb{R}\) be scalars. Then, the vector \(\vec{v}\) given by,
\begin{equation*}
\vec{v} = a_1 \vec{v}_1 + \dots + a_n \vec{v}_n = \sum_{i=1}^n a_i \vec{v}_i
\end{equation*}
is called a linear combination of \(\vec{v}_1, \dots, \vec{v}_n \in V\) with weights \(a_1, \dots, a_n\text{.}\)
Definition 8.1.6.
Let \(\vec{v}_1, \dots, \vec{v}_n \in V\) be vectors. The span of \(\vec{v}_1, \dots, \vec{v}_n\text{,}\) denoted by \(\Span{\vec{v}_1, \dots, \vec{v}_n}\text{,}\) is the set of all linear combinations of \(\vec{v}_1, \dots, \vec{v}_n\text{.}\) In other words,
\begin{equation*}
\Span{\vec{v}_1, \dots, \vec{v}_n} = \set{a_1 \vec{v}_1 + \dots a_n \vec{v}_n : a_1, \dots, a_n \in \mathbb{R}}
\end{equation*}
Theorem 8.1.7. Span is a subspace.
Let \(\vec{v}_1, \dots, \vec{v}_n \in V\text{.}\) Then, \(\Span{\vec{v}_1, \dots, \vec{v}_n}\) is a subspace of \(V\text{.}\)
Then, the set \(\Span{\vec{v}_1, \dots, \vec{v}_n}\) is called the subspace spanned (or generated) by \(\set{\vec{v}_1, \dots, \vec{v}_n}\text{.}\) For any subspace \(H\) of \(V\text{,}\) a set \(\set{\vec{v}_1, \dots, \vec{v}_n}\) of vectors in \(H\) is called a spanning (or generating) set if \(H = \Span{\vec{v}_1, \dots, \vec{v}_n}\text{.}\)
Subsection 8.1.4 Polynomial Spaces
Certain sets of polynomials also act as a vector space. For \(n = 0, 1, 2, \dots\text{,}\) let \(\mathcal{P}_n\) be the set of all polynomials of degree at most \(n\text{,}\)
\begin{equation*}
\mathcal{P}_n = \set{a_0 + a_1 x + a_2 x^2 + \dots + a_n x^n: a_k \in \mathbb{R}}
\end{equation*}
Here, the degree of a polynomial is its highest power of \(x\text{,}\) with a non-zero coefficient. For the polynomial \(p(x) = 0\) with all coefficients equal to 0 (the zero polynomial), its degree is defined to be \(-\infty\) (so it is lower than any \(n\)).
Then, for \(p(x) = a_0 + a_1 x + a_2 x^2 + \dots + a_n x^n, q(x) = b_0 + b_1 x + b_2 x^2 + \dots + b_n x^n\) are elements in \(\mathcal{P}_n\text{,}\) then their sum \(p + q\) is defined by adding corresponding components,
\begin{equation*}
\end{equation*}
For a scalar \(c \in \mathbb{R}\text{,}\) the scalar multiple \(cp\) is given by,
\begin{equation*}
(cp)(x) = c p(x) = ca_0 + (ca_1)x + \dots + (ca_n)x^n
\end{equation*}
In linear algebra, polynomials are thought of as primarily being objects, or expressions, and not as polynomial functions, with an input \(x\) and an ouput \(p(x)\text{.}\)
Polynomial spaces have applications in statistical trend analysis of data.
Note that the set of all polynomials of degree \(n\) is not a vector space, because subtracting two polynomials of degree \(n\) and with the same leading coefficient, results in a polynomial which does not have degree \(n\) (it is has degree less than \(n\)). Thus, the set is not closed under addition.
Subsection 8.1.5 Function Spaces
Let \(V\) be the set of all real-valued functions defined on a set \(D\) (typically, \(D\) is the set of real numbers of some interval on the real line). Two functions are equal if and only if their values are equal for every \(x \in D\text{.}\) For \(f, g \in V\text{,}\) their sum \(f + g\) is defined by \((f + g)(x) = f(x) + g(x)\text{.}\) For a scalar \(c \in \mathbb{R}\text{,}\) the scalar multiple is defined as \((cf)(x) = cf(x)\text{.}\) Then, \(V\) is a vector space. The zero vector is the zero function, the function which is identically zero, i.e. \(f(x) = 0\) for every \(x \in D\text{.}\) The negative of \(f\) is \((-1)f\text{.}\)
