91Ó°ÊÓ

Conjugate symmetry is one of the essential properties of an inner product. It states that the inner product of two vectors should be equal to the conjugate of the inner product taken in the opposite order. Mathematically, this is expressed as:

• \( \langle \mathbf{u}, \mathbf{v} \rangle = \overline{\langle \mathbf{v}, \mathbf{u} \rangle} \)

Since we are working with real numbers in this exercise, taking the conjugate does not change the value, making this requirement look like:

• \( \langle \mathbf{u}, \mathbf{v} \rangle = \langle \mathbf{v}, \mathbf{u} \rangle \)

The given inner product \( \langle \mathbf{u}, \mathbf{v} \rangle = u_1 v_1 \) satisfies this property because of the commutative property of multiplication (i.e., \( u_1 v_1 = v_1 u_1 \)). Thus, conjugate symmetry holds for the specified definition of the inner product. However, it's important to remember that this property doesn't add any new constraints in the real vector space since there's no complex conjugate to worry about.

Linearity in the first argument of the inner product is a foundational principle ensuring that the inner product behaves predictably with respect to addition and scalar multiplication of vectors. This linearity is expressed as:

• \( \langle a\mathbf{u} + b\mathbf{w}, \mathbf{v} \rangle = a\langle \mathbf{u}, \mathbf{v} \rangle + b\langle \mathbf{w}, \mathbf{v} \rangle \)

For the given inner product, we apply this property by evaluating:

• \( \langle a\mathbf{u} + b\mathbf{w}, \mathbf{v} \rangle = (a u_1 + b w_1) v_1 = a u_1 v_1 + b w_1 v_1 \)
• This matches \( a\langle \mathbf{u}, \mathbf{v} \rangle + b\langle \mathbf{w}, \mathbf{v} \rangle \), proving linearity is indeed satisfied in our example.

Linearity assures that the inner product spreads across vector operations in an intuitive way, reflecting how we naturally think about scaling and combining vectors.

Positivity is a crucial property that gives the inner product its geometric meaning, especially in terms of vector lengths and angles. The positivity condition is twofold:

• \( \langle \mathbf{u}, \mathbf{u} \rangle \geq 0 \)
• \( \langle \mathbf{u}, \mathbf{u} \rangle = 0 \Rightarrow \mathbf{u} = \mathbf{0} \)

In this exercise, we find \( \langle \mathbf{u}, \mathbf{u} \rangle = u_1^2 \). Since squares of real numbers are non-negative, the first part of the positivity axiom is satisfied. However, it does not satisfy the second part. For instance, if \( \mathbf{u} = \begin{bmatrix} 0 \ 1 \end{bmatrix} \), then \( \langle \mathbf{u}, \mathbf{u} \rangle = 0 \), even though \( \mathbf{u} eq \mathbf{0} \). This example clearly shows the failure to enforce that the inner product is zero only when the vector is the zero vector. Thus, the positivity axiom does not hold completely for this definition.

The real-valuedness criterion ensures that the inner product of any two vectors yields a real number. This property is typically expressed as:

• \( \langle \mathbf{u}, \mathbf{v} \rangle \in \mathbb{R} \)

For the inner product \( \langle \mathbf{u}, \mathbf{v} \rangle = u_1 v_1 \), each multiplication involves real components from the vectors \( \mathbf{u} \) and \( \mathbf{v} \). Consequently, the result is real since multiplication closes within the real numbers. Thus, the real-valuedness axiom is naturally satisfied. Working in \( \mathbb{R}^2 \), this axiom is straightforward as all vector components and their operations remain within the realm of real numbers. This property assures us that all calculated values via this inner product align with the real-valued systems we frequently encounter in both theoretical and practical applications.