Question

We often write vectors in \(\mathbb{R}^{n}\) as rows. Let \(U\) be a subspace of \(\mathbb{R}^{n},\) and let \(\mathbf{x}\) be a vector in \(\mathbb{R}^{n}\). a. If \(a \mathbf{x}\) is in \(U\) where \(a \neq 0\) is a number, show that \(\mathbf{x}\) is in \(U\) b. If \(\mathbf{y}\) and \(\mathbf{x}+\mathbf{y}\) are in \(U\) where \(\mathbf{y}\) is a vector in \(\mathbb{R}^{n}\), show that \(\mathbf{x}\) is in \(U\)

Short answer

a. \(\mathbf{x}\) is in \(U\); b. \(\mathbf{x}\) is in \(U\).

Step-by-step solution

Understanding Subspace Properties

A subspace of \(\mathbb{R}^{n}\) is a set that satisfies specific properties: it must contain the zero vector, be closed under vector addition, and be closed under scalar multiplication. These properties are essential in solving the problem.

Part (a) Analysis

We are given that \(a \mathbf{x}\) is in \(U\) where \(aeq 0\). Since \(U\) is a subspace, it must be closed under scalar multiplication. Therefore, multiplying by the inverse scalar \(\frac{1}{a}\), we find \(\mathbf{x} = \frac{1}{a}(a \mathbf{x}) \in U\), as \(U\) is closed under multiplication by scalars.

Part (b) Analysis - Given Information

\(\mathbf{y}\) and \(\mathbf{x} + \mathbf{y}\) are elements of \(U\). The property of being closed under addition will be used here.

Using Subspace Closure Under Addition

Subtract \(\mathbf{y}\) from \(\mathbf{x} + \mathbf{y}\). Since both \(\mathbf{x}+\mathbf{y}\) and \(\mathbf{y}\) are in \(U\), and \(U\) is closed under addition, the subtraction \((\mathbf{x} + \mathbf{y}) - \mathbf{y} = \mathbf{x}\) is also in \(U\). This uses the closure property under addition of vectors within the subspace.

Key concepts

Vectors in \(\mathbb{R}^{n}\)

In linear algebra, vectors in \(\mathbb{R}^{n}\) are often written as rows, for example, \( \mathbf{x} = (x_1, x_2, ..., x_n) \). These vectors are elements of the n-dimensional space \(\mathbb{R}^{n}\), a space composed of all possible n-tuples of real numbers.

Each element in a vector represents a coordinate in this space, and these vectors can be added or multiplied by scalars. This flexibility allows for a wide range of operations and transformations, making them foundational in linear algebra, where they are used to model different kinds of data, forces, velocities, and more in various fields.

A valuable insight when dealing with vectors is understanding their geometric interpretation: they can be visualized as arrows in an n-dimensional space, where each component of the vector contributes to the direction and magnitude of the arrow. The ability to navigate freely in \(\mathbb{R}^{n}\) makes vectors versatile in both theoretical and applied scenarios.

Subspaces

Subspaces play a critical role in linear algebra. Specifically, a subspace of \(\mathbb{R}^{n}\) is a subset that follows three important conditions:

• It must contain the zero vector.
• It should be closed under vector addition. This means any two vectors in the subspace can be added together, resulting in another vector that's still within the subspace.
• It must be closed under scalar multiplication, indicating that for any vector in the subspace and any scalar, the product is also in the subspace.

Understanding these conditions is vital when solving problems related to subspaces such as identifying whether certain operations or vectors remain in a given subspace. These properties provide a framework for various proofs and can help determine the structure and limitations of solutions within \(\mathbb{R}^{n}\).

In the context of the given exercise, verifying that multiplying a vector by a scalar or adding two vectors from a subspace results in another vector within the subspace highlights these closure properties and confirms the robustness of subspace characteristics.

Vector addition and scalar multiplication

Vector addition and scalar multiplication are two fundamental operations in linear algebra that support the structure of subspaces.

**Vector Addition**: When you add two vectors, \( \mathbf{u} + \mathbf{v} \), you simply add the corresponding components of the vectors. If both vectors are part of a subspace, their sum must also belong to the subspace. This is called closure under addition. It ensures any combination of vectors within a subspace remains within the subspace.

• Example: For vectors \( \mathbf{u} = (1, 2) \) and \( \mathbf{v} = (3, 4) \), the sum is \( \mathbf{u} + \mathbf{v} = (1+3, 2+4) = (4, 6) \).

**Scalar Multiplication**: This operation involves multiplying a vector by a scalar, \( a \times \mathbf{v} \), affecting both the direction and magnitude of the vector but not its presence in a subspace if the scalar is a real number. Since subspaces are closed under scalar multiplication, multiplying any vector in a subspace by a real number results in another vector in the same subspace.

• Example: For vector \( \mathbf{v} = (3, 4) \) and scalar \( a = 2 \), scalar multiplication gives \( a \times \mathbf{v} = 2 \times (3, 4) = (6, 8) \).

These operations are not just simple calculations but are crucial reasons why vector spaces and subspaces retain their form and functionality in various mathematical and practical applications.