Question

If \(U\) and \(W\) are subspaces, show that \((U+W)^{\perp}=U^{\perp} \cap W^{\perp} .[\) See Exercise 5.1.22.]

Short answer

\((U+W)^{\perp} = U^{\perp} \cap W^{\perp}\) is derived by showing orthogonality conditions.

Step-by-step solution

Understanding the Problem

We are given two subspaces, \(U\) and \(W\), and we need to show that the orthogonal complement of their sum, \((U+W)^{\perp}\), is equal to the intersection of their orthogonal complements, \(U^{\perp} \cap W^{\perp}\).

Definition of Orthogonal Complement

Recall that the orthogonal complement \(V^{\perp}\) of a subspace \(V\) is the set of all vectors that are orthogonal to every vector in \(V\). Hence, \(\mathbf{v} \in (U+W)^{\perp}\) if and only if \(\mathbf{v}\) is orthogonal to every vector in \(U+W\).

Expressing Elements of the Sum of Subspaces

A vector \(\mathbf{u} + \mathbf{w} \) belongs to \(U+W\) if \(\mathbf{u} \in U\) and \(\mathbf{w} \in W\). This implies any vector in \(U+W\) can be expressed as a sum of vectors, one from each subspace.

Deriving Condition for Orthogonality to \(U+W\)

A vector \(\mathbf{v} \) is orthogonal to each vector of the form \(\mathbf{u} + \mathbf{w} \), which means \(\langle \mathbf{v}, \mathbf{u} + \mathbf{w} \rangle = \langle \mathbf{v}, \mathbf{u} \rangle + \langle \mathbf{v}, \mathbf{w} \rangle = 0\) for all \(\mathbf{u} \in U\) and \(\mathbf{w} \in W\).

Two Conditions for Zero Dot Product

The equation \(\langle \mathbf{v}, \mathbf{u} \rangle + \langle \mathbf{v}, \mathbf{w} \rangle = 0\) implies two separate conditions: \(\langle \mathbf{v}, \mathbf{u} \rangle = 0\) for all \(\mathbf{u} \in U\), and \(\langle \mathbf{v}, \mathbf{w} \rangle = 0\) for all \(\mathbf{w} \in W\).

Conclusion by Intersection

Since \(\langle \mathbf{v}, \mathbf{u} \rangle = 0\) implies \(\mathbf{v} \in U^{\perp}\) and \(\langle \mathbf{v}, \mathbf{w} \rangle = 0\) implies \(\mathbf{v} \in W^{\perp}\), the vector \(\mathbf{v}\) must be in both \(U^{\perp}\) and \(W^{\perp}\). Thus, \( (U+W)^{\perp} = U^{\perp} \cap W^{\perp} \).

Key concepts

Subspaces in Linear Algebra

In linear algebra, a subspace is a special set of vectors that follow specific rules. Let's dive into what makes a subspace unique. A subspace must have three main properties:

• Closure under addition: If you take any two vectors from the subspace and add them together, the result stays in the subspace.
• Closure under scalar multiplication: If a vector in the subspace is multiplied by any scalar (just a fancy word for number), the resulting vector stays in the subspace.
• Contains the zero vector: Every subspace must have the zero vector, which is a vector where every component is zero.

Breaking it down, subspaces are like families of vectors. They obey certain rules so when we combine or manipulate them, we don't end up somewhere unexpected. This concept is crucial for many areas, like solving equations or understanding spaces in geometry.

Intersection of Sets

The intersection of sets in mathematics is all about finding common elements between two or more sets. But what does it mean in the context of subspaces? When we consider the intersection of two subspaces, say, \( U \) and \( W \), we're looking for vectors that belong to both \( U \) and \( W \).

Here’s why this is interesting:

• The intersection of subspaces is always a subspace itself. So it follows the same rules as we discussed earlier.
• It helps us understand relationships between different vector spaces and how they overlap.

In the exercise, we see the concept of intersection with the orthogonal complements. The problem asks us to find vectors common to both orthogonal complements of two subspaces. Understanding intersections allows us to figure out really interesting relationships and emphasize how different structures in space connect.

Dot Product

A dot product is a simple yet powerful tool in linear algebra that shows how two vectors relate to each other. When you calculate the dot product of two vectors, it measures how much one vector goes in the direction of the other.

• To find the dot product, take each corresponding component of the vectors, multiply them together and sum it all up.
• The result is a single number, not a vector.

For instance, consider vectors \( \mathbf{a} = [a_1, a_2, a_3] \) and \( \mathbf{b} = [b_1, b_2, b_3] \). The dot product is calculated as \( a_1b_1 + a_2b_2 + a_3b_3 \).

In linear algebra, two vectors are orthogonal if their dot product is zero. This concept is pivotal in the solution, as it provides the condition needed for vectors to be in the orthogonal complement. By examining dot products, we can determine alignment and perpendicularity relationships between vectors, which are fundamental traits when dealing with subspaces and orthogonal complements.