# The Algebraic Backbone of Numerical PDEs: Linear Algebra and Its Challenges
Solving Partial Differential Equations (PDEs) computationally inherently leads to systems of linear algebraic equations, making a strong grasp of Linear Algebra and Numerical Linear Algebra crucial for effectively approximating solutions; key concepts like Gaussian Elimination, ill-conditioned matrices, monotone matrices, and matrix decompositions (e.g., Schur Decomposition) are vital for understanding the properties (sparsity, symmetry, condition number) of these systems and choosing stable, efficient numerical solvers.

> This section, "The Algebraic Backbone of Numerical PDEs: Linear Algebra and Its Challenges," [explores advanced linear algebra concepts crucial for cloud computing](https://viadean.notion.site/The-Algebraic-Backbone-of-Numerical-PDEs-Linear-Algebra-and-Its-Challenges-1f11ae7b9a3280a4beadc79653076130?source=copy_link), including Gaussian Elimination, Ill-conditioned Matrices, Inverse Nonnegative Matrices, Jordan Decomposition, Monotone Matrices, and Schur Decomposition.

## :clapper: Animated result
#### how a continuous mathematical function can be approximated

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