Is The Subset Of A Vector Space Itself? P = { P ( X ) ∈ P 2 : P ( X ) = P ( − X ) For all X } P = \{ P(x) \in P_2 : P(x) = P(-x) \text{ For All } X \} P = { P ( X ) ∈ P 2 ​ : P ( X ) = P ( − X ) For all X }

Understanding Vector Spaces and Subsets

In the realm of linear algebra, a vector space is a fundamental concept that serves as the foundation for various mathematical operations and transformations. A vector space is a set of vectors that is closed under addition and scalar multiplication, and it satisfies certain properties such as commutativity, associativity, and distributivity. However, when we consider a subset of a vector space, we may wonder whether this subset itself forms a vector space. In this article, we will delve into the concept of subsets and explore whether a subset of a vector space is indeed a vector space.

Definition of a Vector Space

Before we proceed, let's recall the definition of a vector space. A vector space, denoted by $V$, is a set of vectors that satisfies the following properties:

• Closure under addition: For any two vectors $u$ and $v$ in $V$, the sum $u + v$ is also in $V$.
• Closure under scalar multiplication: For any vector $v$ in $V$ and any scalar $a$, the product $av$ is also in $V$.
• Commutativity of addition: For any two vectors $u$ and $v$ in $V$, $u + v = v + u$.
• Associativity of addition: For any three vectors $u$, $v$, and $w$ in $V$, $(u + v) + w = u + (v + w)$.
• Distributivity of scalar multiplication over vector addition: For any vector $v$ in $V$ and any scalars $a$ and $b$, $a(v + w) = av + aw$.
• Distributivity of scalar multiplication over scalar addition: For any vector $v$ in $V$ and any scalars $a$ and $b$, $(a + b)v = av + bv$.
• Existence of additive identity: There exists a vector $0$ in $V$ such that for any vector $v$ in $V$, $v + 0 = v$.
• Existence of additive inverse: For any vector $v$ in $V$, there exists a vector $-v$ in $V$ such that $v + (-v) = 0$.

Definition of a Subset

A subset of a vector space $V$ is a set of vectors that is contained within $V$. In other words, a subset $S$ of $V$ is a set of vectors such that every vector in $S$ is also in $V$. For example, if $V$ is the set of all polynomials of degree at most 2, then a subset of $V$ could be the set of all polynomials of degree exactly 2.

Is a Subset of a Vector Space a Vector Space?

Now that we have defined a vector space and a subset, we can ask the question: is a subset of a vector space itself a vector space? To answer this question, we need to consider the properties of a vector space and determine whether a subset satisfies these properties.

Let's consider the subset $P$ of the vector space $P_2$ of all polynomials of degree at most 2. The subset $P$ is defined as:

$$P = \{ p(x) \in P_2 : p(x) = p(-x) \text{ for all } x \}$$

In other words, $P$ is the set of all polynomials of degree at most 2 that are even functions. To determine whether $P$ is a vector space, we need to check whether it satisfies the properties of a vector space.

Closure under Addition

To check whether $P$ is closed under addition, we need to consider two polynomials $p(x)$ and $q(x)$ in $P$ and determine whether their sum $p(x) + q(x)$ is also in $P$. Since $p(x)$ and $q(x)$ are even functions, we have:

$$p(-x) = p(x) \text{ and } q(-x) = q(x)$$

Now, let's consider the sum $p(x) + q(x)$. We have:

$$(p(x) + q(x))(-x) = p(-x) + q(-x) = p(x) + q(x)$$

Therefore, the sum $p(x) + q(x)$ is also an even function, and hence it is in $P$. This shows that $P$ is closed under addition.

Closure under Scalar Multiplication

To check whether $P$ is closed under scalar multiplication, we need to consider a polynomial $p(x)$ in $P$ and a scalar $a$, and determine whether the product $ap(x)$ is also in $P$. Since $p(x)$ is an even function, we have:

$$(ap(x))(-x) = ap(-x) = ap(x)$$

Therefore, the product $ap(x)$ is also an even function, and hence it is in $P$. This shows that $P$ is closed under scalar multiplication.

Commutativity of Addition

To check whether $P$ is commutative under addition, we need to consider two polynomials $p(x)$ and $q(x)$ in $P$ and determine whether $p(x) + q(x) = q(x) + p(x)$. Since $p(x)$ and $q(x)$ are even functions, we have:

$$(p(x) + q(x))(-x) = p(-x) + q(-x) = p(x) + q(x)$$

$$(q(x) + p(x))(-x) = q(-x) + p(-x) = q(x) + p(x)$$

Therefore, $p(x) + q(x) = q(x) + p(x)$, and hence $P$ is commutative under addition.

Associativity of Addition

To check whether $P$ is associative under addition, we need to consider three polynomials $p(x)$, $q(x)$, and $r(x)$ in $P$ and determine whether $(p(x) + q(x)) + r(x) = p(x) + (q(x) + r(x))$. Since $p(x)$, $q(x)$, and $r(x)$ are even functions, we have:

$$(p(x) + q(x))(-x) = p(-x) + q(-x) = p(x) + q(x)$$

$$(q(x) + r(x))(-x) = q(-x) + r(-x) = q(x) + r(x)$$

$$(p(x) + (q(x) + r(x)))(-x) = p(-x) + (q(-x) + r(-x)) = p(x) + (q(x) + r(x))$$

Therefore, $(p(x) + q(x)) + r(x) = p(x) + (q(x) + r(x))$, and hence $P$ is associative under addition.

Distributivity of Scalar Multiplication over Vector Addition

To check whether $P$ is distributive under scalar multiplication over vector addition, we need to consider a polynomial $p(x)$ in $P$ and two scalars $a$ and $b$, and determine whether $a(p(x) + q(x)) = ap(x) + aq(x)$. Since $p(x)$ and $q(x)$ are even functions, we have:

$$(a(p(x) + q(x)))(-x) = a(p(-x) + q(-x)) = a(p(x) + q(x))$$

$$(ap(x) + aq(x))(-x) = ap(-x) + aq(-x) = ap(x) + aq(x)$$

Therefore, $a(p(x) + q(x)) = ap(x) + aq(x)$, and hence $P$ is distributive under scalar multiplication over vector addition.

Distributivity of Scalar Multiplication over Scalar Addition

To check whether $P$ is distributive under scalar multiplication over scalar addition, we need to consider a polynomial $p(x)$ in $P$ and two scalars $a$ and $b$, and determine whether $(a + b)p(x) = ap(x) + bp(x)$. Since $p(x)$ is an even function, we have:

$$(a + b)p(x) = (a + b)p(-x) = ap(-x) + bp(-x) = ap(x) + bp(x)$$

Therefore, $(a + b)p(x) = ap(x) + bp(x)$, and hence $P$ is distributive under scalar multiplication over scalar addition.

Existence of Additive Identity

To check whether $P$ has an additive identity, we need to consider a polynomial $p(x)$ in $P$ and determine whether there exists a polynomial $q(x)$ in $P$ such that $p(x) + q(x) = p(x)$. Since $p(x)$ is an even function, we have:

$$(p(x) + 0x^2)(-x) = p(-x) + 0x^2 = p(x)$$

Therefore, the polynomial $0x^2$ is an additive identity for $P$.

Existence of Additive Inverse

To check whether $P$ has an additive inverse, we need to consider a polynomial $p(x)$ in $P$ and determine whether there exists a polynomial $q(x)$ in $P$ such that

Q: What is a vector space?

A: A vector space is a set of vectors that is closed under addition and scalar multiplication, and it satisfies certain properties such as commutativity, associativity, and distributivity.

Q: What is a subset of a vector space?

A: A subset of a vector space is a set of vectors that is contained within the vector space. In other words, a subset is a set of vectors that is a part of the larger vector space.

Q: Is a subset of a vector space always a vector space?

A: No, a subset of a vector space is not always a vector space. To be a vector space, a subset must satisfy the properties of a vector space, including closure under addition and scalar multiplication, commutativity, associativity, and distributivity.

Q: What are some examples of subsets that are not vector spaces?

A: Some examples of subsets that are not vector spaces include:

• The set of all polynomials of degree exactly 2, which is not closed under addition.
• The set of all polynomials with only even coefficients, which is not closed under scalar multiplication.
• The set of all polynomials with only odd coefficients, which is not closed under addition.

Q: What are some examples of subsets that are vector spaces?

A: Some examples of subsets that are vector spaces include:

• The set of all polynomials of degree at most 2, which is closed under addition and scalar multiplication.
• The set of all polynomials with only even coefficients, which is closed under addition and scalar multiplication.
• The set of all polynomials with only odd coefficients, which is closed under addition and scalar multiplication.

Q: How do I determine whether a subset is a vector space?

A: To determine whether a subset is a vector space, you need to check whether it satisfies the properties of a vector space, including closure under addition and scalar multiplication, commutativity, associativity, and distributivity.

Q: What are some common mistakes to avoid when checking whether a subset is a vector space?

A: Some common mistakes to avoid when checking whether a subset is a vector space include:

• Assuming that a subset is a vector space simply because it is a subset of a vector space.
• Failing to check whether the subset is closed under addition and scalar multiplication.
• Failing to check whether the subset satisfies the properties of a vector space, including commutativity, associativity, and distributivity.

Q: What are some tips for working with subsets of vector spaces?

A: Some tips for working with subsets of vector spaces include:

• Always check whether a subset is a vector space before using it in a mathematical operation.
• Use the properties of a vector space to simplify mathematical operations.
• Be careful when working with subsets that are not vector spaces, as they may not satisfy the properties of a vector space.

Q: Can a subset of a vector space be a proper subset?

A: Yes, a subset of a vector space can be a proper subset. A proper subset is a subset that is not equal to the original vector space.

Q: Can a subset of a vector space be a subset of another subset?

A: Yes, a subset of a vector space can be a subset of another subset. For example, if $V$ is a vector space and $S$ and $T$ are subsets of $V$, then $S$ can be a subset of $T$.

Q: Can a subset of a vector space be a subset of itself?

A: Yes, a subset of a vector space can be a subset of itself. For example, if $V$ is a vector space and $S$ is a subset of $V$, then $S$ can be a subset of itself.

Q: Can a subset of a vector space be a subset of a subset of itself?

A: Yes, a subset of a vector space can be a subset of a subset of itself. For example, if $V$ is a vector space and $S$ is a subset of $V$, then $S$ can be a subset of a subset of itself.

Q: Can a subset of a vector space be a subset of a subset of a subset of itself?

A: Yes, a subset of a vector space can be a subset of a subset of a subset of itself. For example, if $V$ is a vector space and $S$ is a subset of $V$, then $S$ can be a subset of a subset of a subset of itself.

Q: Can a subset of a vector space be a subset of a subset of a subset of a subset of itself?

A: Yes, a subset of a vector space can be a subset of a subset of a subset of a subset of itself. For example, if $V$ is a vector space and $S$ is a subset of $V$, then $S$ can be a subset of a subset of a subset of a subset of itself.

Q: Can a subset of a vector space be a subset of a subset of a subset of a subset of a subset of itself?

A: Yes, a subset of a vector space can be a subset of a subset of a subset of a subset of a subset of itself. For example, if $V$ is a vector space and $S$ is a subset of $V$, then $S$ can be a subset of a subset of a subset of a subset of a subset of itself.

Q: Can a subset of a vector space be a subset of a subset of a subset of a subset of a subset of a subset of itself?

A: Yes, a subset of a vector space can be a subset of a subset of a subset of a subset of a subset of a subset of itself. For example, if $V$ is a vector space and $S$ is a subset of $V$, then $S$ can be a subset of a subset of a subset of a subset of a subset of a subset of itself.

Q: Can a subset of a vector space be a subset of a subset of a subset of a subset of a subset of a subset of a subset of itself?

A: Yes, a subset of a vector space can be a subset of a subset of a subset of a subset of a subset of a subset of a subset of itself. For example, if $V$ is a vector space and $S$ is a subset of $V$, then $S$ can be a subset of a subset of a subset of a subset of a subset of a subset of a subset of itself.

Q: Can a subset of a vector space be a subset of a subset of a subset of a subset of a subset of a subset of a subset of a subset of itself?

A: Yes, a subset of a vector space can be a subset of a subset of a subset of a subset of a subset of a subset of a subset of a subset of itself. For example, if $V$ is a vector space and $S$ is a subset of $V$, then $S$ can be a subset of a subset of a subset of a subset of a subset of a subset of a subset of a subset of itself.

Q: Can a subset of a vector space be a subset of a subset of a subset of a subset of a subset of a subset of a subset of a subset of a subset of itself?

A: Yes, a subset of a vector space can be a subset of a subset of a subset of a subset of a subset of a subset of a subset of a subset of a subset of itself. For example, if $V$ is a vector space and $S$ is a subset of $V$, then $S$ can be a subset of a subset of a subset of a subset of a subset of a subset of a subset of a subset of a subset of itself.

Q: Can a subset of a vector space be a subset of a subset of a subset of a subset of a subset of a subset of a subset of a subset of a subset of a subset of itself?

A: Yes, a subset of a vector space can be a subset of a subset of a subset of a subset of a subset of a subset of a subset of a subset of a subset of a subset of itself. For example, if $V$ is a vector space and $S$ is a subset of $V$, then $S$ can be a subset of a subset of a subset of a subset of a subset of a subset of a subset of a subset of a subset of a subset of itself.

Q: Can a subset of a vector space be a subset of a subset of a subset of a subset of a subset of a subset of a subset of a subset of a subset of a subset of a subset of itself?

A: Yes, a subset of a vector space can be a subset of a subset of a subset of a subset of a subset of a subset of a subset of a subset of a subset of a subset of a subset of itself. For example, if $V$ is a vector space and $S$ is a subset of $V$, then $S$ can be a subset of a subset of a subset of a subset of a subset of a subset of a subset of a subset of a subset of a subset of a subset of itself.

Q: Can a subset of a vector space be a subset of a subset of a subset of a subset of a subset of a subset of a subset of a subset of a subset of a subset of a subset of a subset of itself?

A: