Introduction to Linear Algebra

Hint

This article has little connection with other entries under the "Linear Algebra" category. However, the author believes it is necessary to discuss the essence of linear algebra, trace the origins and connections of concepts, and provide readers with a preliminary yet systematic understanding of linear algebra.

As early as thousands of years ago, ancient people used systems of linear equations to solve problems, and today, linear algebra remains widely applicable.

Linear algebra originates from observation. People discovered that many objects share similar properties, such as:

  • Forces can be decomposed and combined.

  • For any \(k, x_0\), \(k \sin (x - x_0)\) can be decomposed into \(k_1 \sin x + k_2 \cos x\).

These properties are related to the scaling, decomposition, and superposition of the described objects. Linear algebra abstracts these properties from specific objects and studies them as an independent discipline. In OI, knowledge of linear algebra can be directly applied to solve problems or used to optimize algorithms and data structures. For example:

  • Using heavy-light decomposition to maintain a linear basis for finding the maximum XOR sum on a path.

  • Using the matrix tree theorem to transform the problem of counting spanning trees in a graph into computing the determinant of a matrix.

  • Using matrix exponentiation to optimize recurrence relations.

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