Key Sequence

• we defined basis of a vector space—a linearly independent spanning list of that vector space—and shown that to be a basis one has to be able to write a write an unique spanning list
• we show that you can chop a spanning list of a space down to a basis or build a linearly independent list up to a basis
• because of this, you can make a spanning list of finite-dimensional vector spaces and chop it down to a basis: so every finite-dimensional vector space has a basis
• lastly, we can use the fact that you can grow list to basis to show that every subspace of \(V\) is a part of a direct sum equaling to \(V\)

New Definitions

basis and criteria for basis

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