Further Examples of Vector Spaces

Further Examples of Vector Spaces

We will now look at some more examples of vector spaces. We will not verify all ten axioms due to the tedium, however, it is advised that the reader verify that these described sets alongside with their described operations of addition and scalar multiplication satisfy all of the axioms presented on the Vector Spaces page.

The Vector Space of Polynomials of Arbitrary Degree

Let $\wp (\mathbb{F})$ denote the set of all polynomials of arbitrary degree whose coefficients are contained in $\mathbb{F}$. Recall that a polynomial $p(x)$ is generally written in the form $p(x) = a_0 + a_1x + a_2x^2 + a_3x^3 + ...$, where the degree of the polynomial is the largest power in the polynomial, for example $q(x) = 2x^3 + 4x^5$ has degree $5$ and we write $\deg (q) = 5$.

Let $p(x) = a_0 + a_1x + a_2x^2 + a_3x^3 + ...$, $q(x) = b_0 + b_1x + b_2x^2 + b_3x^3 + ...$, and let all the coefficients $a_0, a_1, ..., b_0, b_1, ... \in \mathbb{F}$. This set of vectors $\wp (\mathbb{F})$ alongside with standard polynomial addition $p(x) + q(x) = (a_0 + b_0) + (a_1 + b_1)x + (a_2 + b_2)x^2 + ...$ and scalar multiplication $kp(x) = ka_0 + ka_1x + ka_2x^2 + ...$ forms a vector space.

The Vector Space of Polynomials of Degree ≤ n

Let $\wp _n (\mathbb{F})$ denote the set of all polynomials $p$ such that $\deg ( p ) ≤ n$, in other words, the set of all polynomials whose degree is less than or equal to $n$. We note that this set $\wp _n (\mathbb{F}) \subset \wp ( \mathbb{F})$, however, $\wp _n (\mathbb{F})$ also forms a vector space under the same vector addition and scalar multiplication defined above for polynomials of arbitrary degree.

The Vector Space of Infinite Sequences

Let $F^{\infty}$ denote the set of all infinite sequences $(a_1, a_2, ...)$ whose terms $a_i \in \mathbb{F}$ for all $i \in \mathbb{N}$.

Let $(a_1, a_2, ...) , (b_1, b_2, ...) \in F^{\infty}$ and define vector addition term-wise, that is:

\begin{align} \quad \left ( a_i \right)_{i \in \mathbb{N}} + \left ( b_i \right)_{i \in \mathbb{N}} = (a_1, a_2, ...) + (b_1, b_2, ...) = (a_1 + b_1, a_2 + b_2, ...) = \left ( a_i + b_i\right)_{i \in \mathbb{N}} \end{align}

Let $c \in \mathbb{F}$ and define scalar multiplication to be as follows:

\begin{align} \quad c \left ( a_i \right)_{i \in \mathbb{N}} = c(a_1, a_2, ... ) = (ca_1, ca_2, ...) = \left ( ca_i \right)_{i \in \mathbb{N}} \end{align}

Under these operations the set $F^{\infty}$ forms a vector space.

Note: It is important to note that the entities we are dealing with are sequences are not arbitrary infinite-component vectors despite the notation being the same and despite the set of infinite-component vector also forming a vector space.

The Vector Space of Real Valued Functions

Let $F(-\infty, \infty)$ denote the set of real-valued functions. This set forms a vector space under the operations of function addition, that is if $f, g \in F(-\infty, \infty)$ then $(f + g)(x) = f(x) + g(x)$, and under the operation of scalar multiplication of a function, that is for $a \in \mathbb{F}$, $(af)(x) = af(x)$.

  • From the two formulas above, we can see that $F(-\infty, \infty)$ is closed under addition and scalar multiplication. Furthermore, the zero function, call it $p(x) = 0$ is such that $f + p = f$ is the line that coincides with the $x-axis$. The additive inverse of $f$ is therefore $-f$, that is the function reflected about the $x$-axis. The rest of the axioms can be easily verified.
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