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MUTUALLY ORTHOGONAL FAMILIES OF LINEAR SUDOKU SOLUTIONS JOHN PORCH Abstract. For a class of ?linear? Sudoku solutions, we construct mutually orthogonal families of maximal size for all square orders,

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1. To fill out mutually orthogonal families of, start by identifying a set of vectors or functions. These vectors or functions should be linearly independent and be defined in a common vector space or function space.
2. Next, compute the inner product or dot product between each pair of vectors or functions in the set. The inner product measures the similarity or orthogonality between two vectors or functions.
3. If the inner product between any two vectors or functions is zero, then they are orthogonal to each other. Repeat this process for all possible pairs within the set.
4. Once you have identified a set of mutually orthogonal vectors or functions, you can extend the set by adding additional vectors or functions that are orthogonal to the existing ones. This can be done by adding the condition that the inner product between the new vector or function and all the existing ones is zero.
5. Mutually orthogonal families are commonly used in various applications, such as linear algebra, signal processing, and quantum mechanics. They are particularly useful in orthogonal decomposition, where a vector or function can be represented as a linear combination of orthogonal components.

Who needs mutually orthogonal families of?

1. Researchers and practitioners in the field of linear algebra often require mutually orthogonal families for applications such as solving systems of linear equations, diagonalizing matrices, and finding orthogonal bases.
2. Engineers and scientists working in signal processing utilize mutually orthogonal families in applications like waveform design, channel coding, and multiplexing in communication systems.
3. Physicists and mathematicians employ mutually orthogonal families in quantum mechanics for representing quantum states, calculating probabilities, and analyzing wave functions.

In summary, filling out mutually orthogonal families requires identifying and computing the orthogonality between vectors or functions, while the need for such families spans across various disciplines such as linear algebra, signal processing, and quantum mechanics.

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