Inner Product Space

An inner product space is a vector space equipped with an additional structure called the inner product. The inner product is a mathematical operation that takes two vectors and returns a scalar, providing a way to define geometric concepts like angles and lengths in a vector space.

Formally, an inner product on a vector space $V$ over a field $F$ (either real or complex) is a function: $\langle \cdot, \cdot \rangle : V \times V \to F$ that satisfies the following properties for all vectors $u, v, w \in V$ and scalar $\alpha \in F$:

1. Conjugate Symmetry (or Symmetry for real vector spaces): $\langle u, v \rangle = \overline{\langle v, u \rangle}$ (For real spaces, this simplifies to $\langle u, v \rangle = \langle v, u \rangle$).

2. Linearity in the first argument: $\langle \alpha u + v, w \rangle = \alpha \langle u, w \rangle + \langle v, w \rangle$

3. Positive-Definiteness: $\langle v, v \rangle \geq 0 \quad \text{and} \quad \langle v, v \rangle = 0 \text{ if and only if } v = 0$

Example in Euclidean Space

Consider the vector space $\mathbb{R}^2$ with the standard inner product (also called the dot product), defined as: $\langle u, v \rangle = u_1 v_1 + u_2 v_2$ where $u = (u_1, u_2)$ and $v = (v_1, v_2)$ are two vectors in $\mathbb{R}^2$.

Example Sum:

Let $u = (2, 3)$ and $v = (4, 1)$. We compute the inner product (dot product) as: $\langle u, v \rangle = (2)(4) + (3)(1) = 8 + 3 = 11$

Thus, the inner product of $u$ and $v$ is 11.

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Properties

• Norm: The inner product induces a norm, or length, of a vector: $\| v \| = \sqrt{\langle v, v \rangle}$ For example, for $u = (2, 3)$: $\| u \| = \sqrt{\langle u, u \rangle} = \sqrt{(2)^2 + (3)^2} = \sqrt{13}$

• Angle between vectors: The inner product also allows us to define the cosine of the angle $\theta$ between two vectors: $\cos \theta = \frac{\langle u, v \rangle}{\| u \| \| v \|}$ For the vectors $u = (2, 3)$ and $v = (4, 1)$, this would be: $\cos \theta = \frac{11}{\sqrt{13} \times \sqrt{17}} \approx 0.724$

Would you like more details or have any questions?

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5 Related Questions

1. How does the inner product differ in real and complex vector spaces?
2. How can we generalize inner products to higher-dimensional vector spaces?
3. What is the relationship between the inner product and orthogonality of vectors?
4. How does the concept of inner product extend to function spaces (like $L^2$)?
5. How do you compute the projection of one vector onto another using the inner product?

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Tip:

Always check the properties of the inner product (linearity, symmetry, positive-definiteness) when verifying that a function is indeed an inner product on a vector space.