Use Theorem 4.2 .1 to determine which of the following are...

Use Theorem 4.2 .1 to determine which of the following are subspaces of $P_{3}$ (a) All polynomials $a_{0}+a_{1} x+a_{2} x^{2}+a_{3} x^{3}$ for which $a_{0}=0$. (b) All polynomials $a_{0}+a_{1} x+a_{2} x^{2}+a_{3} x^{3}$ for which $a_{0}+a_{1}+a_{2}+a_{3}=0$ (c) All polynomials of the form $a_{0}+a_{1} x+a_{2} x^{2}+a_{3} x^{3}$ in which $a_{0}, a_{1}, a_{2},$ and $a_{3}$ are rational numbers. (d) All polynomials of the form $a_{0}+a_{1} x,$ where $a_{0}$ and $a_{1}$ are real numbers.

Use Theorem 4.2 .1 to determine which of the following are subspaces of $P_{3}$
(a) All polynomials $a_{0}+a_{1} x+a_{2} x^{2}+a_{3} x^{3}$ for which $a_{0}=0$.
(b) All polynomials $a_{0}+a_{1} x+a_{2} x^{2}+a_{3} x^{3}$ for which $a_{0}+a_{1}+a_{2}+a_{3}=0$
(c) All polynomials of the form $a_{0}+a_{1} x+a_{2} x^{2}+a_{3} x^{3}$ in which $a_{0}, a_{1}, a_{2},$ and $a_{3}$ are rational numbers.
(d) All polynomials of the form $a_{0}+a_{1} x,$ where $a_{0}$ and $a_{1}$ are real numbers.

Key Concepts

Subspace Criterion

In linear algebra, a subset of a vector space is called a subspace if it satisfies three main properties: it is non-empty (usually demonstrated by the inclusion of the zero vector), it is closed under addition (the sum of any two elements in the subset is also in the subset), and it is closed under scalar multiplication (multiplying any element of the subset by a scalar yields an element in the subset). This criterion is essential in determining whether a given subset forms a valid vector space on its own.

Vector Spaces of Polynomials

The collection of polynomials of a certain maximum degree, such as P3 which includes all polynomials of degree 3 or less, forms a vector space. This space is structured with the usual polynomial addition and scalar multiplication, and it provides a natural setting to explore properties like linear combinations and subspaces, governed by algebraic operations similar to those in ??.

Closure Under Addition

To verify that a subset is a subspace, one must check that the sum of any two elements of the subset remains in the subset. This property ensures that the subset is stable under the operation of addition, meaning that combining any two valid 'vectors' (in this context, polynomials) yields another element that still adheres to the subset's defining conditions.

Closure Under Scalar Multiplication

Another crucial part of the subspace test is the requirement that the subset be closed under scalar multiplication. This means if you take any element from the subset and multiply it by any scalar from the field over which the vector space is defined, the resultant element must also be part of the subset. This property guarantees that the scaling operation does not violate any constraints imposed on the subset.

Zero Vector Inclusion

A subset must contain the zero vector to qualify as a subspace, as the zero vector serves as the additive identity in the vector space. The inclusion of the zero vector is not only a necessary condition for the subset to be non-empty, but it also plays a key role in ensuring that the operations of addition and scalar multiplication behave as expected under the subspace structure.

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