21. Linear Algebra
Linear Algebra is fundamental in modern mathematics, especially within Civil Engineering, where it is applied extensively in the analysis of structures, optimization, and numerical simulations. This chapter covers essential concepts of linear algebra such as systems of linear equations, matrix operations, eigenvalues, vector spaces, and their applications in engineering, culminating in advanced topics like diagonalization, the Cayley-Hamilton theorem, and numerical solutions. The interrelation of these concepts provides civil engineers with the tools necessary to tackle complex problems in their field.
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21.12.2Use
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21.17.1Context
What we have learnt
- Linear algebra plays a crucial role in civil engineering applications.
- Understanding systems of equations is foundational for solving engineering problems.
- Matrix operations and eigenvalues are essential for analyzing structural behaviors.
Key Concepts
- -- System of Linear Equations
- A collection of one or more linear equations involving the same set of variables used to solve for unknowns.
- -- Matrix Operations
- Operations involving matrices, including addition, multiplication, and finding determinants, which are crucial for solving linear systems.
- -- Eigenvalues and Eigenvectors
- For a given matrix, eigenvalues are scalars that describe the factor by which the eigenvectors are scaled during a linear transformation.
- -- Rank of a Matrix
- The maximum number of linearly independent row or column vectors in a matrix, indicating the dimension of the vector space generated by its rows or columns.
- -- Linear Transformations
- Functions that map vectors from one vector space to another, preserving the operations of vector addition and scalar multiplication.
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