91Ó°ÊÓ

The non-negativity axiom is a foundational concept for inner products. It states that for any vector \( \mathbf{u} \), the inner product of the vector with itself, \( \langle \mathbf{u}, \mathbf{u} \rangle \), must be greater than or equal to zero. Additionally, it should only be zero if the vector itself is the zero vector. This ensures that distances and lengths, often extracted from inner products, make sense in a geometric way.
Let's consider the given inner product \( \langle \mathbf{u}, \mathbf{v} \rangle = u_1 v_1 - u_2 v_2 \). When we apply this to calculate \( \langle \mathbf{u}, \mathbf{u} \rangle \), we get \( u_1^2 - u_2^2 \). This can result in a negative value, as shown in the example where \( \mathbf{u} = \begin{bmatrix} 1 \ 2 \end{bmatrix} \), leading to \( 1^2 - 2^2 = -3 \), violating the axiom.
This failure implies that the proposed inner product does not sustain the necessary non-negativity condition, thus, isn't a valid inner product in the strict mathematical sense.

The symmetry axiom of an inner product ensures that switching the order of the vectors does not change the result of the inner product. In more formal terms, \( \langle \mathbf{u}, \mathbf{v} \rangle = \langle \mathbf{v}, \mathbf{u} \rangle \).
To verify this with our defined inner product \( \langle \mathbf{u}, \mathbf{v} \rangle = u_1 v_1 - u_2 v_2 \) and \( \langle \mathbf{v}, \mathbf{u} \rangle = v_1 u_1 - v_2 u_2 \), we observe that\:

• \( u_1 v_1 = v_1 u_1 \)
• \( -u_2 v_2 = -v_2 u_2 \)

Thus, the expressions are equal, satisfying the symmetry condition.
This demonstrates that our given inner product is symmetric, aligning with one of the necessary criteria for a valid inner product.

The additivity axiom emphasizes that the inner product is linear concerning addition. It assures us that \( \langle \mathbf{u} + \mathbf{v}, \mathbf{w} \rangle = \langle \mathbf{u}, \mathbf{w} \rangle + \langle \mathbf{v}, \mathbf{w} \rangle \).
In our example, using vectors \( \mathbf{u} = \begin{bmatrix} 1 \ 0 \end{bmatrix} \), \( \mathbf{v} = \begin{bmatrix} 0 \ 1 \end{bmatrix} \), and \( \mathbf{w} = \begin{bmatrix} 1 \ 1 \end{bmatrix} \), we have:

• \( \langle \mathbf{u} + \mathbf{v}, \mathbf{w} \rangle = \langle \begin{bmatrix} 1 \ 1 \end{bmatrix}, \begin{bmatrix} 1 \ 1 \end{bmatrix} \rangle =0 \)
• \( \langle \mathbf{u}, \mathbf{w} \rangle + \langle \mathbf{v}, \mathbf{w} \rangle = 0 \)

Both calculations yield the same result, confirming that additivity is preserved.
This means the additivity axiom holds true for this inner product, illustrating its linear nature over vector addition.

The homogeneity axiom states that the inner product is also linear concerning scalar multiplication of a vector. Mathematically, this implies that \( \langle c\mathbf{u}, \mathbf{v} \rangle = c\langle \mathbf{u}, \mathbf{v} \rangle \).
Using the provided vectors \( \mathbf{u} = \begin{bmatrix} 1 \ 1 \end{bmatrix} \) and \( \mathbf{v} = \begin{bmatrix} 1 \ 1 \end{bmatrix} \), with scalar \( c = 2 \), we compute:

• \( \langle 2\mathbf{u}, \mathbf{v} \rangle = \langle \begin{bmatrix} 2 \ 2 \end{bmatrix}, \begin{bmatrix} 1 \ 1 \end{bmatrix} \rangle = 0 \)
• \( 2 \langle \mathbf{u}, \mathbf{v} \rangle = 0 \)

This equality confirms that the homogeneity axiom holds.
The preservation of this property confirms that scaling a vector does not affect the outcome differently than scaling the result of the inner product directly.