Section 2.15 Preparing to Define Dimension
Discussion Question 2.15.1.
What problems might occur if the dimension of a vector space were defined to be the number of elements in a basis? How is this similar to other situations where the term 'not well defined' is used?
Checkpoint 2.15.2.
Does Theorem 2.11.4 say there is no linearly independent set with more elements than a basis? Explain.
Checkpoint 2.15.3.
Does Theorem 2.13.4 say there is no spanning set with fewer elements than a basis?
Checkpoint 2.15.4.
Find another way to state Theorem 2.15.5.
Theorem 2.15.5.
Considering only finite subsets of \(\mathbb{L}\text{,}\) no linearly independent set has more points than a spanning set.