MHB How Can Basis Vectors Simplify Real-World Problems?

matqkks
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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 'Derivation of equations of stress tensor transformation'
Hello ! I derived equations of stress tensor 2D transformation. Some details: I have plane ABCD in two cases (see top on the pic) and I know tensor components for case 1 only. Only plane ABCD rotate in two cases (top of the picture) but not coordinate system. Coordinate system rotates only on the bottom of picture. I want to obtain expression that connects tensor for case 1 and tensor for case 2. My attempt: Are these equations correct? Is there more easier expression for stress tensor...
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