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Area and volume in Linear Algebra are central concepts that underpin the entire subject, and lead naturally to the rich theory of determinants, a key subject of 18th and 19th century mathematics. This is the fourth lecture of a first course on Linear Algebra, given by N J Wildberger. Here we start with a pictorial treatment of area, then move to an algebraic formulation using bi-vectors. These are two-dimensional versions of vectors introduced in the 1840's by Grassmann. The three dimensional case of volume uses tri-vectors. Video Contents: 0:00 Intro 0:07 Overview 1:05 Area of Fundamental Figures 2:03 Area of a Parallelogram 6:34 Generalizing to an Arbitrary Parallelogram 11:52 Bi-Vectors 15:10 Physical Examples of Bi-Vectors 1 21:12 Physical Examples of Bi-Vectors 2 25:51 Physical Examples of Bi-Vectors 3 28:41 Bi-Vectors in the Plane 1 30:58 Bi-Vectors in the Plane 2 34:03 Unit Bi-Vectors and Area 39:27 Using Bi-Vectors to Compute Area 42:43 Invariance of Ratios of Areas in Affine Geometry 45:51 Three Dimensional Affine Space 49:05 Using Tri-Vectors to Compute Volume 53:23 Exercises ************************ Screenshot PDFs for my videos are available at the website http://wildegg.com. These give you a concise overview of the contents of the lectures for various Playlists: great for review, study and summary. My research papers can be found at my Research Gate page, at https://www.researchgate.net/profile/... My blog is at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things. Online courses will be developed at openlearning.com. The first one, already underway is Algebraic Calculus One at https://www.openlearning.com/courses/... Please join us for an exciting new approach to one of mathematics' most important subjects! If you would like to support these new initiatives for mathematics education and research, please consider becoming a Patron of this Channel at / njwildberger Your support would be much appreciated.