Steps to Verify if a Bilinear Form Satisfies Inner Product Axioms

📚 Understanding Bilinear Forms and Inner Product Axioms

In linear algebra, an inner product is a generalization of the dot product. It's a way to define notions like length, angle, and orthogonality in vector spaces. A bilinear form is a function that takes two vectors as input and returns a scalar, and it's a candidate for being an inner product if it satisfies certain axioms.

📜 Historical Context

The concept of inner products evolved from the study of Euclidean spaces and quadratic forms. Mathematicians like Cauchy and Grassmann laid the groundwork in the 19th century, formalizing these ideas into what we now understand as inner product spaces. The abstraction to general vector spaces came later, providing a powerful tool for analysis and various applications.

🔑 Key Principles

To verify if a bilinear form $\langle \cdot, \cdot \rangle : V \times V \rightarrow \mathbb{F}$ (where $V$ is a vector space over the field $\mathbb{F}$, which is typically the real numbers $\mathbb{R}$ or complex numbers $\mathbb{C}$) satisfies the inner product axioms, you need to check the following properties:

• ➕ Additivity in the First Argument: $\langle u + v, w \rangle = \langle u, w \rangle + \langle v, w \rangle$ for all vectors $u, v, w \in V$.
• 🔢 Homogeneity in the First Argument: $\langle \alpha u, v \rangle = \alpha \langle u, v \rangle$ for all vectors $u, v \in V$ and scalar $\alpha \in \mathbb{F}$.
• 🔄 Conjugate Symmetry: $\langle u, v \rangle = \overline{\langle v, u \rangle}$ for all vectors $u, v \in V$. If the field is $\mathbb{R}$, this simplifies to symmetry: $\langle u, v \rangle = \langle v, u \rangle$.
• ✅ Positive Definiteness: $\langle u, u \rangle \geq 0$ for all vectors $u \in V$, and $\langle u, u \rangle = 0$ if and only if $u = 0$.

🪜 Steps to Verify the Axioms

Here's a step-by-step guide to verifying these axioms:

1. ➕ Verify Additivity:
   • ✏️ Choose arbitrary vectors $u, v, w \in V$.
   • ✍️ Compute $\langle u + v, w \rangle$ using the definition of the bilinear form.
   • ➗ Compute $\langle u, w \rangle + \langle v, w \rangle$ using the definition of the bilinear form.
   • ✔️ Show that $\langle u + v, w \rangle = \langle u, w \rangle + \langle v, w \rangle$.
2. 🔢 Verify Homogeneity:
   • 🧑‍🏫 Choose arbitrary vector $u, v \in V$ and scalar $\alpha \in \mathbb{F}$.
   • 💻 Compute $\langle \alpha u, v \rangle$ using the definition of the bilinear form.
   • 📈 Compute $\alpha \langle u, v \rangle$ using the definition of the bilinear form.
   • ✔️ Show that $\langle \alpha u, v \rangle = \alpha \langle u, v \rangle$.
3. 🔄 Verify Conjugate Symmetry:
   • 🧮 Choose arbitrary vectors $u, v \in V$.
   • 🧭 Compute $\langle u, v \rangle$ using the definition of the bilinear form.
   • ⚛️ Compute $\overline{\langle v, u \rangle}$ using the definition of the bilinear form.
   • ✔️ Show that $\langle u, v \rangle = \overline{\langle v, u \rangle}$. If working with real numbers, show $\langle u, v \rangle = \langle v, u \rangle$.
4. ✅ Verify Positive Definiteness:
   • 🔎 Choose an arbitrary vector $u \in V$.
   • 📊 Compute $\langle u, u \rangle$ using the definition of the bilinear form.
   • ✔️ Show that $\langle u, u \rangle \geq 0$ for all $u$.
   • ✔️ Show that $\langle u, u \rangle = 0$ if and only if $u = 0$.

💡 Real-world Example

Consider the bilinear form on $\mathbb{R}^2$ defined as $\langle u, v \rangle = u_1v_1 + u_2v_2$, where $u = (u_1, u_2)$ and $v = (v_1, v_2)$. Let's verify the inner product axioms:

1. ➕ Additivity: $\langle u + v, w \rangle = (u_1 + v_1)w_1 + (u_2 + v_2)w_2 = u_1w_1 + v_1w_1 + u_2w_2 + v_2w_2 = (u_1w_1 + u_2w_2) + (v_1w_1 + v_2w_2) = \langle u, w \rangle + \langle v, w \rangle$.
2. 🔢 Homogeneity: $\langle \alpha u, v \rangle = (\alpha u_1)v_1 + (\alpha u_2)v_2 = \alpha(u_1v_1 + u_2v_2) = \alpha \langle u, v \rangle$.
3. 🔄 Symmetry: $\langle u, v \rangle = u_1v_1 + u_2v_2 = v_1u_1 + v_2u_2 = \langle v, u \rangle$.
4. ✅ Positive Definiteness: $\langle u, u \rangle = u_1^2 + u_2^2 \geq 0$ for all $u$. If $u_1^2 + u_2^2 = 0$, then $u_1 = 0$ and $u_2 = 0$, so $u = (0, 0)$.

This bilinear form satisfies all inner product axioms, so it is an inner product.

📝 Conclusion

Verifying that a bilinear form satisfies the inner product axioms is crucial for determining whether it can be used to define geometric concepts in a vector space. By systematically checking additivity, homogeneity, conjugate symmetry (or symmetry), and positive definiteness, you can confirm whether a given bilinear form is indeed an inner product. Understanding these steps allows for the effective application of inner product spaces in various fields of mathematics and physics.