Isomorphic Inner Product Spaces | Linear Algebra

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We introduce isomorphic inner product spaces and isomorphisms between inner product spaces. We'll see the definition, and several examples and theorems concerning isomorphic inner product spaces. #linearalgebra

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0:00 Intro
0:59 Definition
2:44 R^n is the Ultimate Lifeform
3:05 Isomorphisms to R^n
4:13 Example of Isomorphic Inner Product Spaces
6:19 Matrix Space Isomorphic to R^n
8:23 Inner Products Preserved
9:09 Conclusion

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If Mn is the space of nx1 column matrices, then v would be nx1, u^T would be 1xn, and u^Tv would just have dimension 1x1. So why is the trace necessary? It would make sense if Mn was the space of 1xn row matrices, am I wrong?

amberyharris