Question

Let \(\mathbf{v}\) denote a nonzero vector in \(\mathbb{R}^{n}\). a. Show that \(P=\left\\{\mathbf{x}\right.\) in \(\left.\mathbb{R}^{n} \mid \mathbf{x} \cdot \mathbf{v}=0\right\\}\) is a subspace of \(\mathbb{R}^{n}\) b. Show that \(\mathbb{R} \mathbf{v}=\\{t \mathbf{v} \mid t\) in \(\mathbb{R}\\}\) is a subspace of \(\mathbb{R}^{n}\) c. Describe \(P\) and \(\mathbb{R} \mathbf{v}\) geometrically when \(n=3\).

Short answer

P is a plane, and \(\mathbb{R}\mathbf{v}\) is a line in \(\mathbb{R}^3\).

Step-by-step solution

Verify Non-Empty Set for P

To show that \( P \) is a subspace, we begin by verifying that it is non-empty. Since \( \mathbf{v} eq \mathbf{0} \), we know that \( \mathbf{0} \cdot \mathbf{v} = 0 \). Therefore, \( \mathbf{0} \) is in \( P \), proving that \( P \) is non-empty.

Closure Under Addition for P

To establish closure under addition, take any \( \mathbf{x}, \mathbf{y} \in P \) such that \( \mathbf{x} \cdot \mathbf{v} = 0 \) and \( \mathbf{y} \cdot \mathbf{v} = 0 \). Then \((\mathbf{x} + \mathbf{y}) \cdot \mathbf{v} = \mathbf{x} \cdot \mathbf{v} + \mathbf{y} \cdot \mathbf{v} = 0 + 0 = 0 \). Thus, \( \mathbf{x} + \mathbf{y} \in P \), showing closure under addition.

Closure Under Scalar Multiplication for P

To prove closure under scalar multiplication, consider any scalar \( c \in \mathbb{R} \) and \( \mathbf{x} \in P \). Then \( (c \mathbf{x}) \cdot \mathbf{v} = c(\mathbf{x} \cdot \mathbf{v}) = c \cdot 0 = 0 \). Thus, \( c \mathbf{x} \in P \), confirming closure under scalar multiplication.

Verify Non-Empty Set for \(\mathbb{R} \mathbf{v}\)

The set \( \mathbb{R} \mathbf{v} = \{t \mathbf{v} \mid t \in \mathbb{R}\} \) is non-empty because it contains the zero vector when \( t = 0 \).

Closure Under Addition for \(\mathbb{R} \mathbf{v}\)

For closure under addition, let \( t \mathbf{v} \) and \( s \mathbf{v} \) be in \( \mathbb{R} \mathbf{v} \). Then \( (t \mathbf{v} + s \mathbf{v}) = (t + s) \mathbf{v} \), which is in \( \mathbb{R} \mathbf{v} \) because \( t + s \) is a real number.

Closure Under Scalar Multiplication for \(\mathbb{R} \mathbf{v}\)

To confirm closure under scalar multiplication, consider a scalar \( c \in \mathbb{R} \) and \( t \mathbf{v} \in \mathbb{R} \mathbf{v} \). Then \( c(t \mathbf{v}) = (ct) \mathbf{v} \), which is in \( \mathbb{R} \mathbf{v} \) because \( ct \) is a real number.

Geometric Description for n=3

For \( n = 3 \), the set \( P \) represents the set of all vectors orthogonal to \( \mathbf{v} \). This forms a plane through the origin in \( \mathbb{R}^3 \). The set \( \mathbb{R} \mathbf{v} \) describes all scalar multiples of \( \mathbf{v} \), which forms a line through the origin parallel to \( \mathbf{v} \) in \( \mathbb{R}^3 \).

Key concepts

Vector Spaces

Vector spaces are foundational in linear algebra that provide a framework for dealing with vectors. A vector space is a collection of vectors that can be added together and scaled by numbers, called scalars, which typically belong to the real numbers \( \mathbb{R} \).

To qualify as a vector space, the set must satisfy certain properties: it must include the zero vector, be closed under vector addition, and be closed under scalar multiplication. Closure under addition means that adding any two vectors in the space results in another vector in the same space. Closure under scalar multiplication means that multiplying any vector by a scalar results in a vector still within the space.

For example, in the provided exercise, the set \( P = \{ \mathbf{x} \in \mathbb{R}^n \mid \mathbf{x} \cdot \mathbf{v} = 0 \} \) is shown to be a vector space by verifying these properties. Similarly, \( \mathbb{R} \mathbf{v} = \{ t \mathbf{v} \mid t \in \mathbb{R} \} \) forms a subspace as it follows these same properties.

Orthogonality

Orthogonality is a concept that refers to two vectors being perpendicular to each other in Euclidean space. In mathematical terms, two vectors \( \mathbf{a} \) and \( \mathbf{b} \) are orthogonal if their dot product equals zero, i.e., \( \mathbf{a} \cdot \mathbf{b} = 0 \).

In the context of the exercise, the set \( P \) is defined as all vectors in \( \mathbb{R}^n \) that are orthogonal to a given vector \( \mathbf{v} \). This property is crucial in linear algebra and geometry because it defines geometric relationships. In three-dimensional space \( \mathbb{R}^3 \), these orthogonal vectors form a plane that passes through the origin and is perpendicular to \( \mathbf{v} \).

Understanding orthogonality is essential, especially when dealing with projections and finding components of vectors that lie along specific directions. It is widely used in various applications such as computer graphics, physics, and engineering.

Scalar Multiplication

Scalar multiplication is an operation involving a vector and a scalar, resulting in a change of the vector's magnitude and possibly its direction. The process involves multiplying each component of the vector by the scalar.

In mathematical notation, for a vector \( \mathbf{v} = (v_1, v_2, \ldots, v_n) \) and a scalar \( c \in \mathbb{R} \), the scalar multiplication results in \( (cv_1, cv_2, \ldots, cv_n) \). This operation is fundamental in showing how vectors scale and transform within a vector space.

In the given exercise, the set \( \mathbb{R} \mathbf{v} = \{ t \mathbf{v} \mid t \in \mathbb{R} \} \) is identified as a subspace because it consists of every possible scalar multiple of \( \mathbf{v} \). This results in a line through the origin in \( \mathbb{R}^3 \), demonstrating how scalar multiplication affects vector orientation and positioning in space.