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When I was learning linear algebra for the first time, I think the hardest concept for me to understand was change of basis, the process of converting one set of basis vectors to another using a linear map. What clicked was when I started viewing change of basis as unit conversions, like converting between Celsius to Fahrenheit.
Basis
Firstly, let's talk about basis in linear algebra, so what is a basis? A basis is just a set of vectors that define your vector space. Every vector in your vector space can be constructed as a linear combination using your basis vectors. In this sense, to really understand a vector space, all you need are your basis vectors, since every other vector is defined as a linear combination of your basis vectors.
For example, using the standard basis (the basis that uses one-hot vectors),
Changing basis
But
Units vs. dimension.