91Ó°ÊÓ

A subspace is a fundamental concept in linear algebra and exists within a larger vector space. To qualify as a subspace, a set must satisfy these three properties:

• The zero vector must be present.
• Closure under addition: If two vectors are in the subspace, their sum must also be in the subspace.
• Closure under scalar multiplication: If a vector is in the subspace, any scalar multiple of that vector must also be in the subspace.

These properties allow subspaces to behave predictably within their vector spaces. Imagine the subspace as a plane within a 3D space, such as \(\mathbb{R}^3\). Every vector operation performed within this plane keeps the results within the subspace.When considering a subspace like \( W \) in \( \mathbb{R}^n \), it can include various vectors. For instance, if \( W \) is spanned by a single vector, every vector in \( W \) can be expressed as multiples of this vector. Understanding subspaces helps us explore orthogonality and vector decomposition in linear algebra.

Orthogonality in linear algebra refers to two vectors being perpendicular to each other. This concept is crucial for simplifying problems and achieving clear geometric interpretations. Two vectors \( \mathbf{a} \) and \( \mathbf{b} \) are orthogonal if their dot product equals zero, expressed as \( \mathbf{a}^T \mathbf{b} = 0 \).In our exercise, orthogonality comes up when decomposing a vector \( \mathbf{v} \) into two components: one belonging to subspace \( W \) and another potentially orthogonal to \( W \). If a vector is orthogonal to every vector in a subspace, it belongs to that subspace's orthogonal complement, denoted as \( W^{\perp} \).Orthogonality helps in vector decomposition, ensuring that components of a vector maintain independence. This independence allows for simplified computations and analysis, such as projecting vectors onto subspaces or calculating distances.

Vector decomposition is the process of breaking down a vector into components that respect certain conditions. In our context, it involves expressing a vector \( \mathbf{v} \) as the sum of two parts: one that lies within a subspace \( W \) and the other that belongs to its orthogonal complement \( W^{\perp} \).The decomposition is commonly represented as: \( \mathbf{v} = \mathbf{w} + \mathbf{w}^{\prime} \), where \( \mathbf{w} \) is in \( W \) and \( \mathbf{w}^{\prime} \) is in \( W^{\perp} \).
For example, if \( W \) is the subspace of \( \mathbb{R}^2 \) spanned by \( (1, 0) \), and we have the vector \( \mathbf{v} = (1, 1) \), we could decompose \( \mathbf{v} \) into \( \mathbf{w} = (1, 0) \) and \( \mathbf{w}^{\prime} = (0, 1) \). \( \mathbf{w}^{\prime} \) is orthogonal to any vector in the span of \( (1, 0) \), solidifying its presence in \( W^{\perp} \).This process is useful in various applications such as projections and simplifying linear equations. By understanding vector decomposition, students can gain insights into more complex vector space manipulations.