- published: 05 Aug 2016
- views: 6659
Linear algebra is the branch of mathematics concerning vector spaces and linear mappings between such spaces. It includes the study of lines, planes, and subspaces, but is also concerned with properties common to all vector spaces.
The set of points with coordinates that satisfy a linear equation forms a hyperplane in an n-dimensional space. The conditions under which a set of n hyperplanes intersect in a single point is an important focus of study in linear algebra. Such an investigation is initially motivated by a system of linear equations containing several unknowns. Such equations are naturally represented using the formalism of matrices and vectors.
Linear algebra is central to both pure and applied mathematics. For instance, abstract algebra arises by relaxing the axioms of a vector space, leading to a number of generalizations. Functional analysis studies the infinite-dimensional version of the theory of vector spaces. Combined with calculus, linear algebra facilitates the solution of linear systems of differential equations.
This introduces the "Essence of linear algebra" series, aimed at animating the geometric intuitions unlerying many of the topics taught in a standard linear algebra course.
Lecture 1: The Geometry of Linear Equations. View the complete course at: http://ocw.mit.edu/18-06S05 License: Creative Commons BY-NC-SA More information at http://ocw.mit.edu/terms More courses at http://ocw.mit.edu
The tools of linear algebra are extremely general, applying not just to the familiar vectors that we picture as arrows in space, but to all sorts of mathematical objects, like functions. This generality is captured with the notion of an abstract vector space. This series comes to you from Khan Academy, watch the rest here: https://www.khanacademy.org/math/linear-algebra/eola-topic/eola/v/eola-preview ------------- Khan Academy offers practice exercises, instructional videos, and a personalized learning dashboard that empower learners to study at their own pace in and outside of the classroom. We tackle math, science, computer programming, history, art history, economics, and more. Our math missions guide learners from kindergarten to calculus using state-of-the-art, adaptive technolog...
A visual understanding of eigenvectors, eigenvalues, and the usefulness of an eigenbasis. This series comes to you from Khan Academy, watch the rest here: https://www.khanacademy.org/math/linear-algebra/eola-topic/eola/v/eola-preview ------------- Khan Academy offers practice exercises, instructional videos, and a personalized learning dashboard that empower learners to study at their own pace in and outside of the classroom. We tackle math, science, computer programming, history, art history, economics, and more. Our math missions guide learners from kindergarten to calculus using state-of-the-art, adaptive technology that identifies strengths and learning gaps. We've also partnered with institutions like NASA, The Museum of Modern Art, The California Academy of Sciences, and MIT to o...
MIT RES.18-009 Learn Differential Equations: Up Close with Gilbert Strang and Cleve Moler, Fall 2015 View the complete course: http://ocw.mit.edu/RES-18-009F15 Instructor: Gilbert Strang A matrix produces four subspaces: column space, row space (same dimension), the space of vectors perpendicular to all rows (the nullspace), and the space of vectors perpendicular to all columns. License: Creative Commons BY-NC-SA More information at http://ocw.mit.edu/terms More courses at http://ocw.mit.edu
Linear algebra studies the dynamics of the simplest possible interactions among multiple variables. Its fundamentals are essential to all areas of mathematics.
How do you translate back and forth between coordinate systems that use different basis vectors? This series comes to you from Khan Academy, watch the rest here: https://www.khanacademy.org/math/linear-algebra/eola-topic/eola/v/eola-preview ------------- Khan Academy offers practice exercises, instructional videos, and a personalized learning dashboard that empower learners to study at their own pace in and outside of the classroom. We tackle math, science, computer programming, history, art history, economics, and more. Our math missions guide learners from kindergarten to calculus using state-of-the-art, adaptive technology that identifies strengths and learning gaps. We've also partnered with institutions like NASA, The Museum of Modern Art, The California Academy of Sciences, and M...
I imagine many viewers are already familiar with vectors in some context, so this video is intended both as a quick review of vector terminology, as well as a chance to make sure we're all on the same page about how specifically to think about vectors in the context of linear algebra.
This video provides a basic outline for how we will go about studying linear algebra by attempting to answer the question: What is Linear Algebra? Sources: The example of a linear system, as well as the terms 'row picture' and 'column picture,' are taken from Gilbert Strang's Introduction to Linear Algebra text and supplemental materials.
The determinant of a linear transformation measures how much areas/volumes change during the transformation. Full playlist: https://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab