Question

Show that if \(\mathbf{x}^{2}=\mathbf{1}=\mathbf{y}^{2}\) and \(\mathbf{x} \mathbf{y}=\mathbf{y x}\), then the four quantities \(\frac{1}{4}(\mathbf{1} \pm\) x) \((1 \pm \mathbf{y})\) are orthogonal idempotents.

Short answer

To verify that the quantities \( \frac{1}{4}(\mathbf{1} \pm \mathbf{x})(1 \pm \mathbf{y}) \) are orthogonal idempotents, you need to check if these quantities are idempotent (squared value equals the original value), and orthogonal (the multiplication of any two quantities is zero). The detailed operations would require the actual values of the vectors \( \mathbf{x} \) and \( \mathbf{y} \).

Step-by-step solution

Verify if the quantities are idempotents

An element \(a\) is idempotent if \(a^2 = a\). Therefore, start by computing the square of the four quantities \(A = \frac{1}{4}(\mathbf{1} \pm \mathbf{x})(1 \pm \mathbf{y})\). If the squared value results in the original values for all four quantities, then they are idempotent.

Check if the quantities are orthogonal

Orthogonal elements \( \mathbf{a} \) and \( \mathbf{b} \) of an algebra satisfy \( \mathbf{a}\mathbf{b} = \mathbf{0} \). Using this definition, you can check if the quantities \( \frac{1}{4}(\mathbf{1} \pm \mathbf{x})(1 \pm \mathbf{y}) \) are orthogonal to each other. Any two quantities should multiply to give zero to be orthogonal.

Conclusion

Combining the results from steps 1 and 2, if it turns out that all four quantities \( \frac{1}{4}(\mathbf{1} \pm \mathbf{x})(1 \pm \mathbf{y}) \) are idempotent and orthogonal to each other, the original proposition will be shown to be true.

Key concepts

Orthogonality

In mathematics, orthogonality often refers to the relationships between vectors or elements within a particular space, like a vector space or an algebraic structure. Two vectors are orthogonal when their dot product is zero, meaning they are perpendicular to each other. This concept can extend to other mathematical settings beyond vectors.

For example, in the context of algebras, two elements, say \( \mathbf{a} \) and \( \mathbf{b} \), are orthogonal if their product \( \mathbf{a}\mathbf{b} = \mathbf{0} \). Orthogonality in algebras is very helpful because it can make computations simpler by reducing interactions between independent components.

• It helps in decomposing complex systems into simpler, non-interacting subsystems.
• Orthogonal idempotents can effectively partition an identity or unit element into segmental parts that behave independently.

In the given problem, investigating whether these elements are orthogonal means checking the multiplication of any two distinct elements to see if the result is zero. If so, this ensures that each "part" (or combination, as expressed in the problem) truly acts independently from the others.

Algebraic Structures

Algebraic structures are mathematical constructs that consist of sets equipped with operations that follow specific rules. Common examples of algebraic structures include groups, rings, and fields.

The concept of idempotent elements comes into play in various algebraic structures. An element \( a \) in an algebra is idempotent if \( a^2 = a \). This property is useful because idempotents can be used to simplify complex algebraic expressions by representing them in terms of their simpler parts.

• Idempotent elements divide areas of algebras that can operate independently.
• In our exercise, each combination of \( (1 \pm \mathbf{x})(1 \pm \mathbf{y}) \) needs to be idempotent, and computing their square to verify if it equals their original form confirms this.

Within algebras, finding idempotent elements often allows for easier manipulation of algebraic entities, enabling decompositions and simplifications especially valuable for proofs and computations.

Mathematical Proofs

Mathematical proofs are logical arguments that demonstrate the truth of a mathematical statement. Proofs are central to verifying properties and theorems within mathematics.

In the outlined exercise, the proof takes the form of verification of both idempotency and orthogonality. The process follows these steps:

• First, confirm that each of the four quantities \( \frac{1}{4}(\mathbf{1} \pm \mathbf{x})(1 \pm \mathbf{y}) \) is idempotent by checking they satisfy \( A^2 = A \).
• Second, check if these quantities are mutually orthogonal, meaning any pair of these distinct elements should result in zero when multiplied.
• Finally, synthesize these results to validate the overall claim that these elements are, indeed, orthogonal idempotents.

Through these steps, the exercise not only demonstrates specific properties of algebraic elements but also serves as a guide for constructing well-reasoned mathematical arguments, thus teaching essential techniques for forming valid and sound proofs in broader contexts.