Axler 3.D

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

• invertability
• isomorphism + isomorphic vector spaces
• matrix of a vector
• operator

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)
  • “each column of a matrix represents where each of the basis of the input gets taken to”
  • So applying a vector to a matrix shows the linear combination of what where the basis sent
• 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

Interesting Factoids