Linear Algebra: Matrix Algebra, Determinants, & Eigenvectors from Coursera

This course is the second course in the Linear Algebra Specialization. In this course, we continue to develop the techniques and theory to study matrices as special linear transformations (functions) on vectors. In particular, we develop techniques to manipulate matrices algebraically. This will allow us to better analyze and solve systems of linear equations. Furthermore, the definitions and theorems presented in the course allow use to identify the properties of an invertible matrix, identify relevant subspaces in R^n,

We then focus on the geometry of the matrix transformation by studying the eigenvalues and eigenvectors of matrices. These numbers are useful for both pure and applied concepts in mathematics, data science, machine learning, artificial intelligence, and dynamical systems. We will see an application of Markov Chains and the Google PageRank Algorithm at the end of the course.

What's inside

Syllabus

Matrix Algebra

In this module, we now look at what arithmetic operation we can perform on nxm matrices and how these operations correspond to operations on functions. In particular, we will view at matrix multiplication AB as a composition of function A(B(x)). In this way, algebraic properties like non-commutativity will become more apparent. We will also look for those matrices that are invertible. Since we no longer have the Horizontal Line Test, new tests for invertibility will be needed. This will lead to the study of the very important matrix invariant, the determinant.

Subspaces

In this module we investigate the structure of R^n by formally defining the notion of a subspace. These special sets are those that look like smaller versions of R^n that pass through the origin. These subsets have invariants called a dimension which captures a notion of size. The linear algebra definition of dimension, which uses the notion of linearly independent vectors, matches our intuition in low dimensions where lines have dimension one and planes have dimension two. These sets, and their sizes, turn out to be another tool to student matrices as functions as both the zeros and image of a matrix are subspaces of R^n.

Determinants

The determinant is a real number calculated from a square matrix that determines the invertibility of a square matrix. Its value characterizes the invertibility of the matrix. The determinant also has a geometric meaning: the absolute value of the determinant scales the volumes of sets under the function. In this module, we will show how to calculate the determinant of nxn matrices and study its properties.

Eigenvectors and Eigenvalues

In this module we study special vectors, called eigenvectors, of a linear transformation defined by a square matrix A. These are vectors whose image is easily visualized as they are scaled by a real number called the eigenvalue. While eigenvalues can be complex numbers, we do not consider that case in this course. Eigenvalues and eigenvectors are central to the theory of discrete dynamical systems, differential equations, and Markov chains and the eigentheory presented here also appear in settings in more advanced pure math courses.

Diagonalization and Linear Transformations

In this module we continue our study of eigenvalues and eigenvectors, in particular how they relate to diagonalizable matrices. Eigenvectors are so important: they make understanding linear transformations easy. They are the "axes" (directions) along which a linear transformation acts simply by "stretching/compressing" and/or "flipping"; eigenvalues give you the factors by which this compression occurs. The more directions you have along which you understand the behavior of a linear transformation, the easier it is to understand the linear transformation; so you want to have as many linearly independent eigenvectors as possible associated to a single linear transformation.

Final Assessment

Congratulations on reaching the final assessment! Review all vocabulary and theorems before attempting the final quiz below. Think about what each theorem is saying both algebraically as well as geometrically. Provide examples (with pictures in R^2 and R^3) along with counterexamples of each theorem and vocabulary term. Lastly, be sure to work through some examples for computation, looking for any of the shortcuts in the calculations when possible. In addition, there is an optional project that applies the theory of this course. You will see how eigenvalues and eigenvectors are applied to Markov Chains and the Google Page Rank algorithm. I strongly recommend you attempt this project. Good luck!

Good to know

Know what's good

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Develops advanced techniques and theory to study matrices as linear transformations

Applies eigenvalues and eigenvectors to Markov Chains and the Google PageRank algorithm

Offers an interactive learning experience with a mix of media

Teaches linear algebra concepts that are relevant in data science, machine learning, and artificial intelligence

Provides a deeper understanding of matrix transformations through the study of eigenvalues and eigenvectors

Activities

Be better prepared before your course. Deepen your understanding during and after it. Supplement your coursework and achieve mastery of the topics covered in Linear Algebra: Matrix Algebra, Determinants, & Eigenvectors with these activities:

Practice Matrix Algebra Operations

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Review the basic operations of matrix algebra to strengthen your understanding of matrices as linear transformations.

Browse courses on Matrix Multiplication

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Review the definitions of matrix addition, subtraction, and multiplication.
Practice performing these operations on small matrices (e.g., 2x2 and 3x3).
Verify the properties of matrix operations, such as associativity and distributivity.

Solve Systems of Linear Equations using Matrices

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Apply matrix operations to solve systems of linear equations efficiently and accurately.

Browse courses on Systems of Linear Equations

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Set up the system of equations as a matrix equation.
Perform matrix operations to transform the matrix into an upper triangular form.
Use back substitution to solve for the variables.

Attend a Linear Algebra Workshop

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Enhance your understanding of linear algebra concepts through interactive workshops led by experts in the field.

Browse courses on Linear Algebra

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Find a workshop on linear algebra that aligns with your interests.
Register for the workshop and attend all sessions.
Participate actively in discussions and ask questions to clarify concepts.

Five other activities

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Explore Geometric Properties of Matrix Transformations

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Understand the geometric significance of eigenvalues and eigenvectors in describing the behavior of matrix transformations.

Browse courses on Eigenvalues and Eigenvectors

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Follow a guided tutorial on the concepts of eigenvalues and eigenvectors.
Visualize the geometric transformations associated with different types of eigenvalues and eigenvectors.
Apply these concepts to analyze the behavior of real-world systems.

Compile Resources on Matrix Theory

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Expand your knowledge and understanding of matrix theory by gathering and organizing relevant resources.

Browse courses on Linear Algebra

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Search for books, articles, and online resources on matrix theory.
Organize the resources by topic or difficulty level.
Share the compilation with other students or interested individuals.

Identify Subspaces and Dimensionality in Matrices

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Develop a deeper understanding of subspaces and dimensionality by creating a visual representation of these concepts in matrices.

Browse courses on Subspaces

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Identify subspaces within a given matrix.
Calculate the dimensionality of these subspaces.
Create a visual representation, such as a graph or diagram, to illustrate the subspaces and their dimensionality.

Tutor Students in Linear Algebra

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Reinforce your understanding of linear algebra by helping others learn the subject and develop their problem-solving skills.

Browse courses on Linear Algebra

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Offer your services as a tutor to students in need of assistance with linear algebra.
Prepare lesson plans and materials to effectively convey concepts.
Provide individualized support and guidance to students.

Participate in a Linear Algebra Challenge

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Challenge yourself and test your understanding of linear algebra concepts by participating in a competition.

Browse courses on Linear Algebra

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Find an online or local linear algebra challenge.
Solve the problems independently or as part of a team.
Analyze your results and identify areas for improvement.

Career center

Learners who complete Linear Algebra: Matrix Algebra, Determinants, & Eigenvectors will develop knowledge and skills that may be useful to these careers:

Machine Learning Engineer

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Data Scientist

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Quantitative Analyst

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Operations Research Analyst

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Actuary

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Statistician

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Financial Analyst

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Market Researcher

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Software Engineer

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Computer Scientist

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Economist

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Systems Engineer

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Research Scientist

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Data Analyst

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