isomorphisms. Somebody’s new favourite word since last year.

## Key Sequence

- we showed that a linear map’s inverse is unique, and so named the inverse \(T^{-1}\)
- we then showed an important result, that injectivity and surjectivity implies invertability
- this property allowed us to use invertable maps to define isomorphic spaces, naming the invertable map between them as the isomorphism
- we see that having the same dimension is enough to show invertability (IFF), because we can use basis of domain to map the basis of one space to another
- we then use that property to establish that matricies and linear maps have an isomorphism between them: namely, the matrixify operator \(\mathcal{M}\).
- this isomorphism allow us to show that the dimension of a set of Linear Maps is the product of the dimensions of their domain and codomain (that \(\dim \mathcal{L}(V,W) = (\dim V)(\dim W)\))

- We then, for some unknown reason, decided that right this second we gotta define matrix of a vector, and that linear map applications are like matrix multiplication because of it. Not sure how this relates
- finally, we defined a Linear Map from a space to itself as an operator
- we finally show an important result that, despite not being true for infinite-demensional vector space, injectivity is surjectivity in finite-dimensional operators

## New Definitions

## Results and Their Proofs

- linear map inverse is unique
- injectivity and surjectivity implies invertability
- two vector spaces are isomorphic IFF they have the same dimension
- matricies and Linear Maps from the right dimensions are isomorphic
- \(\dim \mathcal{L}(V,W) = (\dim V)(\dim W)\)
- \(\mathcal{M}(T)_{.,k} = \mathcal{M}(Tv_{k})\), a result of how everything is defined (see matrix of a vector)
- linear maps are like matrix multiplication
- injectivity is surjectivity in finite-dimensional operators

## Questions for Jana

~~why doesn’t axler just say the “basis of domain” directly (i.e. he did a lin comb instead) for the second direction for the two vector spaces are isomorphic IFF they have the same dimension proof?~~because the next steps for spanning (surjectivity) and linear independence (injectivity) is made more obvious~~clarify the matricies and Linear Maps from the right dimensions are isomorphic proof~~~~what is the “multiplication by \(x^{2}\)” operator?~~literally multiplying by \(x^{2}\)~~how does the matrix of a vector detour relate to the content before and after? I suppose an isomorphism exists but it isn’t explicitly used in the linear maps are like matrix multiplication proof, which is the whole point~~because we needed to close the loop of being able to linear algebra with matricies completely, which we didn’t know without the isomorphism between matricies and maps