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You can absolutely visualize why [1, 0] and [-1, 0] is not a basis for R^2! The problem is if you plot them in R^2, then they and the origin all lie on a single
by obastani 8y ago
You can absolutely visualize why [1, 0] and [-1, 0] is not a basis for R^2! The problem is if you plot them in R^2, then they and the origin all lie on a single line. No matter how you sum copies of these vectors, you can never "escape" this line. Thus, you cannot construct vectors in R^2 that do not lie on this line.
- platform 8y agoYes, that's exactly how the author of the article is explaining it (including the picture). I was just commenting that it was good that, the author took time to explain this. And those kinds of nuances/counter examples are not always available in explanatory dynamic visualization tools (such as the ones referenced in the comments here [1] ). To be clear, I am not arguing against dynamic visualizations. These are very nice, very very helpful, and very time consuming to build and present on a website. I am just suggesting that showing negative examples is a complimentary, and powerful explanation technique. [1] http://setosa.io/ev/eigenvectors-and-eigenvalues/ http://setosa.io/ev/eigenvectors-and-eigenvalues/