Question

If \(U\) is a subspace of \(\mathbb{R}^{n}\), show that \(\operatorname{proj}_{U} \mathbf{x}=\mathbf{x}\) for all \(\mathbf{x}\) in \(U\)

Short answer

If \(\mathbf{x} \in U\), then \(\operatorname{proj}_{U} \mathbf{x} = \mathbf{x}\).

Step-by-step solution

Define Projection

The projection of a vector \(\mathbf{x}\) onto a subspace \(U\) is defined as the vector \(\operatorname{proj}_{U} \mathbf{x}\) that is in \(U\) and is the closest point in \(U\) to \(\mathbf{x}\). This means that the difference \(\mathbf{x} - \operatorname{proj}_{U} \mathbf{x}\) is orthogonal to \(U\). This ensures \(\operatorname{proj}_{U} \mathbf{x}\) achieves the minimum distance from \(\mathbf{x}\) to \(U\).

Assume \(\mathbf{x} \in U\)

Let \(\mathbf{x} \in U\), which means that \(\mathbf{x}\) is already a vector in the subspace \(U\). By the properties of subspaces, \(\mathbf{x}\) should remain unchanged when projected.

State the Orthogonality Condition

Since \(\mathbf{x} \in U\), the difference between \(\mathbf{x}\) and any point within \(U\) is zero, because \(\mathbf{x}\) is already within \(U\). Therefore, \(\mathbf{x} - \operatorname{proj}_{U} \mathbf{x} = \mathbf{0}\), which vacuously satisfies the orthogonality condition; this confirms that \(\mathbf{x} = \operatorname{proj}_{U} \mathbf{x}\).

Conclude the Proof

Hence, for any vector \(\mathbf{x}\) in the subspace \(U\), projecting \(\mathbf{x}\) onto \(U\) results in \(\operatorname{proj}_{U} \mathbf{x} = \mathbf{x}\), as \(\mathbf{x}\) is by definition already the nearest point to itself within \(U\). This establishes our claim.

Key concepts

Subspaces in Linear Algebra

Subspaces are fundamental constructs in linear algebra. Think of a subspace as a smaller universe within a vector space, like a room inside a larger house. To qualify as a subspace of a vector space, a set must satisfy three conditions:

• It must contain the zero vector, ensuring closure under addition and scalar multiplication.
• Any linear combination of vectors in the subspace must also be in the subspace. This is what we mean by "closure."
• The set of vectors must be non-empty and closed, meaning that the addition of any two vectors in the set remains in the set.

For example, if you consider all vectors lying in a plane through the origin in three-dimensional space, that set forms a subspace. When dealing with subspaces such as \(U\) of \(\mathbb{R}^{n}\), understanding these principles helps explain why projecting a vector within \(U\) keeps it unchanged.

Orthogonality

Orthogonality is a concept describing the idea of vectors being "perpendicular" to each other, making them meet at right angles. When you project a vector onto a subspace, the resulting projection has the smallest possible "shadow" or difference when measured, by being orthogonal to the subspace.
This means that the residual vector, \(\mathbf{x} - \operatorname{proj}_{U} \mathbf{x}\), is orthogonal to every vector in the subspace \(U\). This principle is crucial for optimizing distances, as it ensures no component of the difference vector lies within the subspace.

Orthogonality plays a vital role in efficiently managing space, particularly in computational applications, because it allows for separating components of vectors into independent directions. When \(\mathbf{x}\) is already within \(U\), the difference is zero, fulfilling the orthogonality condition automatically by the nature of being a part of the subspace.

Vector Spaces

A vector space extends the concept of a quantitative geometric environment where vectors freely "live," interact, and combine. These are mathematical structures over a field (like real numbers \(\mathbb{R}\)), defined by certain operations: vector addition and scalar multiplication.
Every vector space is defined by axioms such as associativity, commutativity of addition, having additive identity (zero vector), and additive inverses. Scalar multiplication must be distributive and associative, and there should be a multiplicative identity for scalars.

• Vectors are elements in these spaces, represented as ordered lists of numbers, like coordinates in geometry. \(\mathbf{x} = (x_1, x_2, \, ...\,, x_n)\) is a vector in \(\mathbb{R}^n\), such as a point in n-dimensional space.
• Subspaces are vector spaces themselves, confined within larger vector spaces.

In any vector space, understanding these operations allows for analyzing various vector behaviors, such as finding projections. Knowing that \(\mathbf{x}\) resides in vector subspace \(U\) explains why the projection operation leaves it unchanged, thus maintaining essential structural properties of the space.