Master Vector Equations - Part One | Explanations and Examples

Learn all about the fundamentals of Euclidean space and the role of vectors in linear algebra. In this video, you’ll discover the key features of Euclidean space, starting with vectors in R^2—how to write their components, perform scalar multiplication, and add or subtract vectors both algebraically and graphically. We'll also dive into vectors in R^n and their algebraic properties. Next, we'll cover linear combinations of vectors, working through several examples to determine if a vector can be expressed as a combination of other vectors. Finally, we'll apply these concepts to solve a real-world problem by calculating the center of mass of a thin rectangular plate.

Be sure to download the free guided notes (link below) to follow along and practice on your own. These notes include space to highlight key concepts and solve the example problems as we work through them.

*Link to Guided Notes*
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_Linear Equations in Linear Algebra:_

_Matrix Algebra:_

_Determinants_

_Vector Spaces_

_Eigenvalues and Eigenvectors_

_Orthogonality and Least Squares_

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*Timestamps*
00:00 Introduction
00:30 Euclidean Space
03:24 Vectors in R^2
10:44 Vectors in R^3
11:33 Vectors in R^n
12:07 Algebraic Properties of R^n
13:43 Linear Combinations
23:47 Application Problems

*More About Understand The Math*
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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