91Ó°ÊÓ

Orthogonal vectors are pivotal in the concept of inner product spaces. When two vectors, say \( \mathbf{w} \) and \( \mathbf{u} \), are orthogonal, it means they are at a right angle to each other. In mathematical terms, this relationship is expressed as their inner product equalling zero: \( \langle \mathbf{w}, \mathbf{u} \rangle = 0 \). This signifies that the two vectors do not influence each other in any linear combination, making orthogonality analogous to being independent in certain spaces.

In the context of the problem, \( \mathbf{w} \) is orthogonal to both \( \mathbf{u} \) and \( \mathbf{v} \). As such, any vector that is a combination of \( \mathbf{u} \) and \( \mathbf{v} \) will also be orthogonal to \( \mathbf{w} \). This is because the contribution of each component vector, whose inner products with \( \mathbf{w} \) are zero, results in the net inner product of the combination with \( \mathbf{w} \) being zero as well.

Linear combinations are a fundamental concept in vector spaces. When we talk about a linear combination, we refer to creating a new vector by adding together scaled versions of other vectors. For instance, given vectors \( \mathbf{u} \) and \( \mathbf{v} \) and scalars \( r \) and \( s \), a linear combination is expressed as \( r \mathbf{u} + s \mathbf{v} \).

This new vector formed can hold crucial information about the space defined by \( \mathbf{u} \) and \( \mathbf{v} \). It spreads the influence of both vectors proportionally, depending on \( r \) and \( s \).

In our problem, we're distilling the component of \( \mathbf{w} \)'s orthogonality through the linear combination of \( \mathbf{u} \) and \( \mathbf{v} \). By demonstrating that \( \langle \mathbf{w}, r\mathbf{u} + s\mathbf{v} \rangle = 0 \), we're showing that the resulting vector formed through their linear combination remains orthogonal to \( \mathbf{w} \).

Inner products offer a mathematical way to measure the angle and length relationship between two vectors. There are critical properties of inner products, which include linearity, symmetry, and positivity.

Linearity in the first argument means that for any scalar \( a \), and vectors \( \mathbf{x} \), \( \mathbf{y} \), and \( \mathbf{z} \), the inner product satisfies \( \langle a\mathbf{x} + \mathbf{y}, \mathbf{z} \rangle = a \langle \mathbf{x}, \mathbf{z} \rangle + \langle \mathbf{y}, \mathbf{z} \rangle \). This allows us to simplify expressions when dealing with linear combinations. Symmetry suggests that \( \langle \mathbf{x}, \mathbf{y} \rangle = \langle \mathbf{y}, \mathbf{x} \rangle \), highlighting that the order of the vectors does not affect the inner product value.

Positivity states that \( \langle \mathbf{x}, \mathbf{x} \rangle \geq 0 \) for any vector \( \mathbf{x} \), which basically means that a vector always has a non-negative length squared. The zero value implies the vector is actually a zero vector.

These properties are instrumental when proving orthogonality in inner product spaces. In our case, using linearity, we can express \( \langle \mathbf{w}, r \mathbf{u} + s \mathbf{v} \rangle \) as \( r \langle \mathbf{w}, \mathbf{u} \rangle + s \langle \mathbf{w}, \mathbf{v} \rangle \), which simplifies to zero if \( \mathbf{w} \) is orthogonal to both \( \mathbf{u} \) and \( \mathbf{v} \). This confirms that the linear combination \( r\mathbf{u} + s\mathbf{v} \) is orthogonal to \( \mathbf{w} \) as well.