Question

In each case, determine which of axioms \(\mathrm{P} 1-\mathrm{P} 5\) fail to hold. a. \(V=\mathbb{R}^{2},\left\langle\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right)\right\rangle=x_{1} y_{1} x_{2} y_{2}\) b. \(V=\mathbb{R}^{3}\) \(\left\langle\left(x_{1}, x_{2}, x_{3}\right),\left(y_{1}, y_{2}, y_{3}\right)\right\rangle=x_{1} y_{1}-x_{2} y_{2}+x_{3} y_{3}\) c. \(V=\mathbb{C},\langle z, w\rangle=z \bar{w}\), where \(\bar{w}\) is complex conjugation d. \(V=\mathbf{P}_{3},\langle p(x), q(x)\rangle=p(1) q(1)\) e. \(V=\mathbf{M}_{22},\langle A, B\rangle=\operatorname{det}(A B)\) f. \(V=\mathbf{F}[0,1],\langle f, g\rangle=f(1) g(0)+f(0) g(1)\)

Short answer

a) Fails P3, P4, P5. b) Fails P5. c) Valid. d) Fails P3. e) Fails P3, P4, P5. f) Fails P3, P5.

Step-by-step solution

Analyze part a

We are given that the inner product \( \langle (x_1, y_1), (x_2, y_2) \rangle = x_1 y_1 x_2 y_2 \), which suggests a product of four factors. An inner product on \( \mathbb{R}^2 \) should take the form \( x_1 x_2 + y_1 y_2 \). This proposed product is not linear; it violates the linearity axiom over the first and second components, specifically, \( P3 \) and \( P4 \). It also may not be positive definite \( P5 \), since \( \langle (1, 0), (0, 1) \rangle = 0 \), but non-zero vectors should not yield zero.

Analyze part b

The inner product given is \( \langle (x_1, x_2, x_3), (y_1, y_2, y_3) \rangle = x_1 y_1 - x_2 y_2 + x_3 y_3 \). This product is linear, respects symmetry \( P1 \), and is positive definite by definition when checking simple cases. However, it isn't guaranteed to be non-negative for all non-zero vectors, which violates \( P5 \).

Analyze part c

In this case, \( \langle z, w \rangle = z \bar{w} \) is given. For complex spaces, this meets \( P1 \), \( P2 \), and \( P4 \), because of properties of conjugation. However, \( \langle z, z \rangle = z \bar{z} = |z|^2 \) is validly positive, respecting \( P5 \). This being a standard dot product, it satisfies all axioms.

Analyze part d

Here, \( \langle p(x), q(x) \rangle = p(1) q(1) \) is proposed. This definition only evaluates the polynomials at one point, which is not enough to ensure linearity over the whole vector space \( \mathbf{P}_3 \). Also, this violates the linearity condition \( P3 \), making the space non-standard as an inner product space.

Analyze part e

We have \( \langle A, B \rangle = \det(AB) \) for matrices. The product \( \det(AB) \) does not ensure linearity \( P3, P4 \) because determinants have a different nature of interaction with scalar multiplication and addition. Furthermore, it cannot guarantee positivity, potentially violating \( P5 \) as well because non-zero matrices can have zero determinant.

Analyze part f

The function space has inner product \( \langle f, g \rangle = f(1)g(0) + f(0)g(1) \). This operation doesn't consistently distribute over the vector space, failing linearity \( P3 \) and potentially symmetry \( P1 \). It is particularly problematic for ensuring positivity \( P5 \) due to the independent evaluation at endpoints only.

Key concepts

Vector Spaces

Imagine a space where not only points, but entire vectors are the main players. That's a vector space! These sets of vectors are subjected to two main operations: addition and scalar multiplication. This creates a structure filled with vectors that can stretch, shrink, and add together according to specific rules.

Within this space, each vector follows specific axioms, known as vector space axioms. There are ten of these, including attributes like the existence of a zero vector and distributive properties. Vector spaces can exist in various forms, like \(\mathbb{R}^2\), \(\mathbb{R}^3\), and even the space of polynomials.

Understanding vector spaces is essential because they provide the groundwork for understanding inner products and the nature of various mathematical functions.

Linearity Axioms

Linearity plays a pivotal role in the world of mathematics, especially when working with vectors. The linearity axioms are fundamental to defining inner products and include two properties: linearity in the first component (additivity and homogeneity) and linearity in the second component.

In simpler terms, these axioms dictate how we can 'mix' and 'scale' components in a vector space. If an inner product satisfies these properties, it ensures that combining or scaling vectors within the space behaves predictably. This is why, for example, checking that a product like \(\langle (x_1, y_1), (x_2, y_2) \rangle = x_1 y_1 x_2 y_2 \) violates linearity is crucial. Understanding and verifying linearity is fundamental in asserting the behavior and characteristics of vector operations.

Positive Definite

The idea of being positive definite is another core concept in understanding inner products. An inner product space is positive definite if it satisfies certain positivity conditions, notably that for any non-zero vector \(v\), the inner product \(\langle v, v \rangle\) is greater than zero.

This concept ensures that our mathematical structure has a meaningful geometry where distances and angles are always non-negative. For instance, in some of the original exercises, we recognized that \(\langle (x_1, y_1), (x_2, y_2) \rangle = x_1 y_1 x_2 y_2 \) might not be positive definite because results can be zero even when non-zero vectors are involved.

Emphasizing the positive definiteness of an inner product helps in ensuring that concepts such as lengths and norms consistently behave as expected.

Symmetry in Mathematics

Symmetry in mathematics often implies that operations can be swapped without changing the result. For inner products, this symmetry property is defined by \(\langle u, v \rangle = \langle v, u \rangle\), which means the operation is unchanged when the vectors switch places.

This property brings a sense of balance and predictability to the structure of a vector space. It allows certain processes such as calculating angles between vectors to be consistent, no matter the order of vectors involved.

In our exercises, exploring this symmetry was essential. For instance, confirming whether certain inner products like \(\langle A, B \rangle = \det(AB)\) exhibit this symmetry helps in understanding how well they define a vector space. Symmetry is fundamental in identifying if an operation properly qualifies as an inner product.