Linear Algebra

Research and review topics in linear algebra. In these articles and videos, I begin with matrices, linear maps, vector spaces, and then continue.

Linear Algebra is an undergraduate or a graduate mathematics course. The undergraduate version often emphasizes applications and applying theorems, and the emphasis is usually placed on matrices, whereas linear maps are the graduate course’s focal point. Topics that undergraduate students learn in this course are solving linear systems of equations, vector spaces, bases, and dimensions. After that, linear transformations and matrices, kernel and image, orthogonality, determinants, and various applications. In the graduate approach, instructors expect students to prove theorems, and the exercises are usually in the form of proving statements.

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Systems of Linear Equations (and System Equivalency) [Video]

I begin with 2 by 2 and 3 by 3 linear systems of equations. I then discuss consistent and inconsistent systems of linear equations and work through several examples. The general system of linear equations is defined, and parametrizing a solution set of a linear system is demonstrated. I explain augmented matrices, row operations, and when two linear systems of equations are called equivalent.

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Eigenvalues and Eigenvectors (Find and Use Them)

Are you looking for more information on eigenvalues and eigenvectors? In this article, I work through several examples and go over the characteristic equation, and then I go over even more examples. All theorems are provided with proof.

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The Determinant of a Matrix (Theory and Examples)

Okay, so there is the definition of a determinant, and there are ways of computing the determinant. The definition of determinant involves patterns and a signature, and this approach is useful in proofs. But to determine the determinant in practice, it’s helpful to have expansion by cofactor. In this article, I detail both.

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Inner Products and Orthonormal Bases

You have been familiar with the Cartesian plane for a while, and so you know about angles. How do we work with angles in a vector-space? In this article, I go over inner-product spaces and work through the Cauchy-Schwarz Theorem and the Triangular Theorem. I also discuss orthonormal bases.

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Gram-Schmidt Process and QR Factorization

In this article, I explain what the Gram-Schmidt Process is and what a QR Factorization is. I do this by working through several examples. Also, I include the commutative diagrams for the corresponding transformations.

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Orthogonal Matrix and Orthogonal Projection Matrix

In this article, I cover orthogonal transformations in detail. After that, I present orthogonal and transpose properties and orthogonal matrices. Towards the end, I examine the orthogonal projection matrix and provide many examples and exercises.

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Orthonormal Bases and Orthogonal Projections

In this article, I begin with the fact that orthonormal vectors are linearly independent and thus form a basis for the subspaces generated by them. After that, I explore the orthogonal projection and properties of the orthogonal complement. Towards the end, I detail the Pythagorean Theorem, the Cauchy-Schwarz Theorem, the Law of Cosines, and the Triangular Inequality Theorem.

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Coordinates (Vectors and Similar Matrices)

Are you trying to understand why similar matrices are essential? Why are they used? This article covers coordinate vectors, coordinate space, and the matrix representation of a linear transformation. After that, I discuss similar matrices and their importance.

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Subspaces and Linear Independence

In this article, I give an elementary introduction to subspaces. After that, I motivate the concepts of linear independence, spanning set, and basis. Then, I prove that a basis of the subspace gives a unique representation. Towards the end, I explore the dimension of a subspace. You will find many examples and exercises.

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Linear Independence and Bases

In this article, I cover linear independence and the Linear Independence Lemma. After that, I discuss bases and finite-dimensional vector spaces. Several examples and corollaries are presented.

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The Kernel of a Matrix (and Image)

I discuss the kernel of a linear transformation and its basic properties. After that, I discuss the image of a linear transformation and its basic properties. Then, I investigate the Rank-Nullity Theorem, which combines the dimension of the image space (rank) and the dimension of the kernel space (nullity) into a single beautiful equation.

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Introduction to Vectors and Subspaces

Not familiar 100% with what a vector space is? In this article, I cover vectors spaces and some basic theorems. I also emphasize linear combinations, the unions and intersections of subspaces, and provide several exercises.

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Vector Spaces and Basic Theorems

Do you know what a field is? In this article, I go over some basic properties of a vector space over a field. These theorems are good exercises for someone, just learning how to write proofs.

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Infinite Dimensional Vector Space

A vector space that is not finite-dimensional is called an infinite-dimensional vector space. Can you think of any off-hand? In this article, I give an example, and I prove that every vector space must have a basis.

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Finite-Dimensional Linear Spaces

Do you know what linearly independent means? Do you know what a spanning set is? A vector space with a finite linearly independent set of spanning vectors is called a finite-dimensional linear space. In this article, I go over the basic theorems for these spaces by providing their proofs.