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Linear Algebra for Differential Equations

Apr 22, 2025

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Linear Algebra Crash Course for Systems of Differential Equations

Vectors

  • Definition: A vector is a directed line segment starting at the origin, terminating at the specified coordinates.
  • Notation: Represented by a lowercase letter with an arrow (e.g., ( \vec{x} )).
  • Components: Typically considered with x and y components, but can be generalized.
  • Equivalent Vectors: Vectors with the same direction and magnitude are identical, regardless of their position.
  • Vector Operations:
    • Addition: Add corresponding components.
    • Scalar Multiplication: Multiply each component by the scalar (e.g., ( 3 \times (1, 2) = (3, 6) )).
    • Magnitude: Calculated using the Pythagorean theorem: ( |\vec{x}| = \sqrt{a^2 + b^2} ).
    • Functions as Scalars: Vectors can be scaled by functions, not just constants.

Matrices

  • Definition: A collection of vectors.
  • Notation: Rows by columns (e.g., a 2x2 matrix has 2 rows and 2 columns).
  • Matrix Operations:
    • Addition: Add corresponding elements.
    • Scalar Multiplication: Multiply each element by the scalar.
    • Matrix Multiplication: Defined by scaling the columns of the first matrix by the elements of the second (resulting vector's size is determined by rows and columns compatibility).

Systems of Equations

  • Can be represented using matrices.
  • Matrix Equations: Convert between matrix equation and scalar form.
  • Coefficient Matrix: Contains the coefficients of the system's variables.

Differential Equations in Matrix Form

  • Write systems like ( \frac{dx}{dt} = 3x + 2y ) and ( \frac{dy}{dt} = 2x - 4y ) as matrices.
  • Important for representing and solving differential systems.

Eigenvectors and Eigenvalues

  • Eigenvectors: Special vectors that, when multiplied by a matrix, result in a vector that is a scaled version of the original (no rotation).
  • Eigenvalues: Scalars (denoted by ( \lambda )) by which the eigenvector is scaled.
  • Characteristics:
    • Square matrices have eigenvectors and eigenvalues.
    • Eigenvectors are not unique but maintain constant ratios.
    • Eigenvalues are unique for a given matrix.

Calculus with Vectors

  • Derivatives and Integrals: Calculated component-wise.

Practical Application

  • Transformation: Multiplying a matrix by a vector transforms it (can scale or rotate it).
  • Geometric Interpretation: Visualize how vectors change under matrix multiplication.

Summary

  • Eigenvectors and eigenvalues are essential in analyzing matrices, especially for solving differential equations.
  • Tools like Wolfram Alpha can be used for calculation.

Next Steps

  • Application of these concepts to solving differential equations will be covered in the next lesson.

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