91Ó°ÊÓ

When we speak about polynomial space, particularly in this context, we are referring to collections of polynomials that share certain properties, such as degree. Here, the polynomials are of the form \( p(t) = at^2 \), meaning they belong to the space of polynomials of degree 2, denoted as \( \mathbb{P}_2 \). This space includes all quadratic polynomials where the highest power of \( t \) is 2. Each polynomial in this space is uniquely defined by a single real coefficient \( a \), which multiplies \( t^2 \).

Polynomial spaces like \( \mathbb{P}_2 \) are important because they help in understanding and manipulating polynomials using vector space concepts. Essentially, they provide a framework where polynomials can be added together or multiplied by scalars, much like ordinary vectors.

One of the essential properties of a subspace is closure under addition. This means that if you take two polynomials from the set and add them, the result is also within the set. Let's illustrate this with the polynomials \( p_1(t) = a t^2 \) and \( p_2(t) = b t^2 \).

When you add these polynomials together, you get:

• \( p_1(t) + p_2(t) = at^2 + bt^2 = (a + b)t^2 \)

The result, \( (a+b)t^2 \), is still a polynomial of the form \( at^2 \), meaning it also belongs to our set. Since adding any two polynomials results in another polynomial of the same form, the set is closed under addition. This closure property is crucial for the set to qualify as a subspace.

Another critical property for determining if a set is a subspace is closure under scalar multiplication. This involves taking any polynomial from our set and multiplying it by a scalar from the real numbers to see if the result remains in the set.

Suppose you have a polynomial \( p(t) = at^2 \) and a real number \( c \.\) Multiply them to get:

• \( c(p(t)) = c(at^2) = (ca)t^2 \)

Here, \( ca \) is just another real number, which means \( (ca)t^2 \) is of the same form as the original polynomial \( at^2 \). So, multiplying by any scalar produces another polynomial within the set. This assures us that our set is closed under scalar multiplication, reinforcing its status as a subspace.

A fundamental requirement for any subspace is that it must include the zero polynomial. This is the polynomial that evaluates to zero for any input, and it serves as the "zero vector" in vector space terms.

In our polynomial set, the zero polynomial occurs when the coefficient \( a = 0 \). Substituting \( a = 0 \) into \( p(t) = at^2 \), we get:

• \( p(t) = 0 \cdot t^2 = 0 \)

This confirms the presence of the zero polynomial in the set. Including the zero polynomial ensures that the set satisfies all necessary conditions to be a subspace of \( \mathbb{P}_2 \), confirming a complete and perfect polynomial subspace.