Master Vector Spaces and Subspaces: Definitions, Rules, and Examples

Learn about the fundamental concepts of vector spaces and subspaces in linear algebra. Starting with the definition of a vector space, we delve into key examples such as Euclidean spaces, matrix spaces, and polynomial spaces. We cover the ten rules that define a vector space, including vector addition and scalar multiplication, and walk through steps to determine whether a set forms a vector space. Additionally, we define subspaces, explain their properties, and show examples, including the null space, column space, and the set of triangular matrices. Finally, we provide a step-by-step guide on how to verify if a set is a subspace, along with illustrative examples. Perfect for engineering students or anyone looking to solidify their understanding of linear algebra!

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*Timestamps*
00:00 Introduction
00:32 Definition and Examples of Vector Spaces
01:35 Ten Axioms or Rules of Vector Spaces
04:26 Steps for Determining if a Set is a Vector Space and Examples
08:50 Definition and Examples of Subspaces
11:05 Examples of Determining if a Set is a Subspace

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Understand The Math provides students and lifelong learners with access to free, straightforward, and engaging mathematics instruction. I’m Cheryl Hile, the founder of Understand The Math, and I’m passionate about ensuring that accurate and engaging mathematics instruction is available to everyone! I have a Ph.D. in Engineering Science and Applied Mathematics and have over 25 years of teaching experience at the university level. This channel provides videos that contain clear explanations, step-by-step problem solving strategies, and carefully worked out example problems. In the video descriptions, you'll find a link to free guided notes. Students can use these notes to highlight concepts and fill in answers to example problems while watching the videos, and later, they can review what they've learned.

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