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 EquationsFree

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Reduction of SystemsFree

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Rank

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Classification of Solutions

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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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Your status is based on your weighted accuracy which accounts for the difficulty of the questions.

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Re-answering questions correctly will improve your weighted average status.

Linear Combinations

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Linear Independence

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Spanning Sets

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Bases

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Coordinates

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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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Accuracy is based on your most recent attempt.

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Your status is based on your weighted accuracy which accounts for the difficulty of the questions.

Your weighted accuracy is based on your most recent attempts compared to everyone else’s first attempts.

Re-answering questions correctly will improve your weighted average status.

Matrix Algebra

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Inverses and Invertible Matrices

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Special Matrices

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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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Accuracy is based on your most recent attempt.

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Your status is based on your weighted accuracy which accounts for the difficulty of the questions.

Your weighted accuracy is based on your most recent attempts compared to everyone else’s first attempts.

Re-answering questions correctly will improve your weighted average status.

Basic Transformations

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Visualizing Linear Transformations

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Change of Basis

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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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Accuracy is based on your most recent attempt.

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Your status is based on your weighted accuracy which accounts for the difficulty of the questions.

Your weighted accuracy is based on your most recent attempts compared to everyone else’s first attempts.

Re-answering questions correctly will improve your weighted average status.

Determinants

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Eigenvectors and Eigenvalues

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Diagonalization

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Single Value Decomposition

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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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Accuracy is based on your most recent attempt.

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Your status is based on your weighted accuracy which accounts for the difficulty of the questions.

Your weighted accuracy is based on your most recent attempts compared to everyone else’s first attempts.

Re-answering questions correctly will improve your weighted average status.

Subspaces of R^n

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Abstract Vector Spaces

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Linear Transformations Between Vector Spaces

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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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Accuracy is based on your most recent attempt.

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Your status is based on your weighted accuracy which accounts for the difficulty of the questions.

Your weighted accuracy is based on your most recent attempts compared to everyone else’s first attempts.

Re-answering questions correctly will improve your weighted average status.

Visualizing Vector Operations

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Dot Product

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General Inner Product 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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Accuracy is based on your most recent attempt.

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Your status is based on your weighted accuracy which accounts for the difficulty of the questions.

Your weighted accuracy is based on your most recent attempts compared to everyone else’s first attempts.

Re-answering questions correctly will improve your weighted average status.

Orthogonality

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Orthogonal vectors

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Orthogonal sets and bases

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Orthonormal sets and bases

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Orthogonal matrices

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Gram-Schmidt

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Orthogonal Projection

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