Key Sequence

• Because Length of Basis Doesn’t Depend on Basis, we defined dimension as the same, shared length of basis in a vector space
• We shown that lists of the right length (i.e. dim that space) that is either spanning or linearly independent must be a basis—“half is good enough” theorems
• we also shown that \(dim(U_1+U_2) = dim(U_1)+dim(U_2) - dim(U_1 \cap U_2)\): dimension of sums

New Definitions

• dimension

Results and Their Proofs

• Length of Basis Doesn’t Depend on Basis
• lists of right length are basis
  • linearly independent list of length dim V are a basis of V
  • spanning list of length of dim V are a basis of V
• dimension of sums

Questions for Jana

• Example 2.41: why is it that \(\dim U \neq 4\)? We only know that \(\dim \mathcal{P}_{3}(\mathbb{R}) = 4\), and \(\dim U \leq 4\). Is it because \(U\) (i.e. basis of \(U\) doesn’t span the polynomial) is strictly a subset of \(\mathcal{P}_{3}(\mathbb{R})\), so there must be some extension needed? because we know that \(U\) isn’t all of \(\mathcal{P}_{3}\).

Interesting Factoids

OMGOMGOMG its Linear Maps time! “One of the key definitions in linear algebra.”

Key Sequence

• We define these new-fangled functions called Linear Maps, which obey \(T(u+v) = Tu+Tv\) and \(T(\lambda v) = \lambda Tv\)
• We show that the set of all linear maps between two vector spaces \(V,W\) is denoted \(\mathcal{L}(V,W)\); and, in fact, by defining addition and scalar multiplication of Linear Maps in the way you’d expect, \(\mathcal{L}(V,W)\) is a vector space!
  • this also means that we can use effectively the \(0v=0\) proof to show that linear maps take \(0\) to \(0\)
• we show that Linear Maps can be defined uniquely by where it takes the basis of a vector space; in fact, there exists a Linear Map to take the basis anywhere you want to go!
• though this doesn’t usually make sense, we call the “composition” operation on Linear Maps their “product” and show that this product is associative, distributive, and has an identity

New Definitions

• Linear Map — additivity (adding “distributes”) and homogeneity (scalar multiplication “factors”)
  • \(\mathcal{L}(V,W)\)
  • any polynomial map from Fn to Fm is a linear map
  • addition and scalar multiplication on \(\mathcal{L}(V,W)\); and, as a bonus, \(\mathcal{L}(V,W)\) a vector space!
  • naturally (almost by the same \(0v=0\) proof), linear maps take \(0\) to \(0\)
• Product of Linear Maps is just composition. These operations are:
  • associative
  • distributive
  • has an identity

Results and Their Proofs

• technically a result: any polynomial map from Fn to Fm is a linear map
• basis of domain of linear maps uniquely determines them

Questions for Jana

• why does the second part of the basis of domain proof make it unique?

Key Sequence

• we defined the null space and injectivity
  • from that, we showed that injectivity IFF implies that null space is \(\{0\}\), essentially because if \(T0=0\) already, there cannot be another one that also is taken to \(0\) in an injective function
• we defined range and surjectivity
• we showed that these concepts are strongly related by the fundamental theorem of linear maps: if \(T \in \mathcal{L}(V,W)\), then \(\dim V = \dim null\ T + \dim range\ T\)
• from the fundamental theorem, we showed the somewhat intuitive pair about the sizes of maps: map to smaller space is not injective, map to bigger space is not surjective
• we then applied that result to show results about homogeneous systems
  • homogenous system with more variables than equations has nonzero solutions
  • inhomogenous system with more equations than variables has no solutions for an arbitrary set of constants

New Definitions

• null space
  • injectivity
• range
  • surjectivity
• homogeneous system

Results and Their Proofs

• the null space is a subspace of the domain
• injectivity IFF implies that null space is \(\{0\}\)
• the fundamental theorem of linear maps
• “sizes” of maps
  • map to smaller space is not injective
  • map to bigger space is not surjective
• solving systems of equations:
  • homogenous system with more variables than equations has nonzero solutions
  • inhomogenous system with more equations than variables has no solutions for an arbitrary set of constants

Questions for Jana

• “To prove the inclusion in the other direction, suppose v 2 null T.” for 3.16; what is the first direction? maybe nothing maps to \(0\)

matricies!!!!

Key Sequence

• matricies exist, you can add them, scalarly multiply them, and actually multiply them
• they can represent Linear Maps by showing where they take basis
• unsurprisingly, the set of matricies of a shape is a vector space

New Definitions

• matricies
  • matrix of Linear Map
  • matrix addition and scalar multiplications
  • matrix multiplication
• \(\mathbb{F}^{m,n}\)

Results and Their Proofs

• sums and scalar multiplication of matricies, and why they work to represent Linear Maps
• \(\mathbb{F}^{m,n}\) is a vector space

Interesting Factoids

its literally matricies

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