#source Axler2.C
#ref #disorganized #incomplete
1 #definition dimension
The dimension of \(V\) (denoted \(\text{dim }V\) is the length of a basis in \(V\) - This relies on Axler2.35: Basis length does not depend on the basis Any two bases of a finite-dimensional vector space have the same length
1.1 Results
1.2 Axler2.38 Dimension of a subspace
If \(V\) is finite-dimensional and \(U\) is a subspace of \(V\), then \(\text{dim }U \le \text{dim }V\) - Intuitive. The basis of a subspace is shorter than the basis of the original vecspace, because otherwise it's span would be larger than the original vecspace (because bases are linearly independent + len lin-indep \(\le\) len span).
1.3 Axler2.39 Linearly independent list of the right length is a basis
All linearly independent lists of the length \(\text{dim }V\) are bases. - Intuitive. If it's already linearly independent meaning each element brings "new information", then if there's that many elements then there should be that much information.
1.4 Axler2.43 Dimension of a sum
If \(U_1\) and \(U_2\) are subspaces of a finite dimensional vecspace, then \[\text{dim}(U_1+U_2) = \text{dim }U_1 + \text{dim }U_2 - \text{dim}(U_1\cap U_2)\] - This inducts into something analogous to PIE! KBrefPrincipleInclusionExclusion