# Linear Algebra

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## Linear Equations and Vectors

As the study of vector spaces and linear transformations, linear algebra is a fundamental mathematical tool. In this introduction, explore the basic tools of vectors, vector spaces, matrices, and linear equations.

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## Linear Independence, Spanning, and Bases

Linear independence helps to eliminate redundant vectors and efficiently define a set for a particular vector space. By manipulating vectors through combinations, learn about how to determine a basis and what implications that has on a vector space.

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## Matrices

Matrices can act as the "shorthand" of linear transformations. Establish a foundation in the basics of matrices and how to manipulate them. Extend your knowledge to special kinds of matrices and matrix factorization.

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## Linear Transformations

One of the most powerful concepts in linear algebra, linear transformations create a map from one vector space to another. Explore basic transformations and useful ways of visualizing them. Then determine cases in which the basis can be changed.

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## Determinants and Eigenvectors

Eigenvectors are special vectors that do not change direction under linear transformations. They are associated with eigenvalues which describe their scalar transformation. Learn to use determinants to find eigenvalues and then apply eigenvectors themselves.

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## Vector Spaces

Begin discovering the nuances of vector spaces beyond basic linear transformations. Explore the concept of subspaces, specifically of $\mathbb{R}^n$, and more abstract vector spaces. Utilize these concepts to practice more advanced linear transformations.

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## Inner Product Spaces

Inner product spaces are unique vector spaces in which pairs of vectors are associated by a scalar quantity. Learn about the dot product in this theme and how it applies to general vector spaces.

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## Orthogonality

The property of being perpendicular in linear algebra is known as orthogonality. Learn how to determine the orthogonality of vectors, including the Gram-Schmidt method, and apply orthogonality to projections.

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