Chapter 2C Reading

• how should we define dimension?
  • should make the dimension of \(F^n=n\)
  • nice to define as the length of a basis
    • all basis of a given vector space have the same length

basis len does not on basis
-> any two bases of a finite-dimension vector space have the same length

cool proof!

• V is finite dimensional
• \(B_1\) and \(B_2\) both bases of \(V\).
• Thus, \(B_1\) is linearly independent in \(V\) and \(B_2\) spans \(V\)
  • by @axler-2.23 (len of lin indepedent list \(\leq\) len of spanning list)
  • len(\(B_1\)) \(\leq\) len(\(B_2\))
  • swap \(B_1\) and \(B_2\)
    • len(\(B_2\)) \(\leq\) len(\(B_1\))
      
    • therefore, \(B_1\) and \(B_2\) need to be of equal len.

title: dimension
the *dimension* of a finite-dimensional vector space is the len of any basis of the vector space.
denoted as dim $V$ if V is finite-dimensional

• which are all the same!

title: dimension of a subspace @2.38
if $V$ is finite-dimensional and $U$ is a subspace of $V$, then dim $U\leq$ dim $V$.

• linearly independent list of the right length is a basis

title: linearly independant list of the right length is a basis @2.39
suppose $V$ if finite-dimensional. Then very linearly indepedent list of vectors in $V$ with length dim $V$ is a basis $V$

• makes sense, because linearly independent list with the len of the dimension has to include all the info, and thus it spans, and thus it is a basis.
  • we don't need to check that it spans!
    • instead, we just need to check that it is linearly independent and that it has the same len as the dim
• why does the list being linearly independent mean that its dim is >= 3? #question @axler-2.41
• what??? #question what is happening at the end of @axler-2.41

title: spanning list of the right length is a basis @2.42
suppose $V$ is finite-dimensional. Then every spanning list of vectors in $V$ with length dim $V$ is a basis of $V$.

• every list of vecs which both spans and is the same len as the dim is a basis.
  • because in order to span, you need all the info
  • and to be the len of the dim, you can't have repeat info
  • so it's a basis.

title: dimension of a sum @2.43
if $U_1$ and $U_2$ are subspaces of a finite-dimensional vector space, then 
$$dim(U_1+U_2) = dim U_1 +dim U_2 - dim(U_1\cap U_2)$$