MHB How Can Basis Vectors Simplify Real-World Problems?

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Basis vectors provide a foundational framework for understanding vector spaces and coordinate systems, crucial for solving real-world problems. Introducing them through relatable applications can engage students more effectively than traditional definitions. By framing basis vectors as tools for uniquely identifying points in space, educators can highlight their practical significance. This approach emphasizes the importance of coordinate systems in various fields, enhancing student motivation. Understanding basis vectors ultimately simplifies complex concepts and fosters problem-solving skills.
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What is attractive way to introduce basis vectors? I am looking for a hook that students will find motivating. It needs to have an impact. I have normally introduced it by just stating independent vectors that span the space.
 
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I've never taught linear algebra, but selecting a basis is essentially selecting a coordinate system, which allows identifying each point (vector) with a unique sequence of numbers. Coordinate systems are obviously crucial for solving many practical problems.
 
Thread 'How to define a vector field?'

Thread 'How to define a vector field?'

Hello! In one book I saw that function ##V## of 3 variables ##V_x, V_y, V_z## (vector field in 3D) can be decomposed in a Taylor series without higher-order terms (partial derivative of second power and higher) at point ##(0,0,0)## such way: I think so: higher-order terms can be neglected because partial derivative of second power and higher are equal to 0. Is this true? And how to define vector field correctly for this case? (In the book I found nothing and my attempt was wrong...
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