Macroeconomics Exam Review 22

Macroeconomics Exam Review 22 - which is also linearly...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: which is also linearly independent ( since x +1 / ∈ lin ). If lin +1 = , then +1 is a basis and we are done. Otherwise, there exists some x +2 ∈ ∖ lin +1 . Adding x +2 to +1 gives a new set of + 2 elements +2 = +1 ∪ { x +2 } which is also linearly independent ( since x +2 / ∈ lin +2 ). Repeating this process, we can construct a sequence of linearly independent sets , +1 , +2 ... such that lin ⫋ lin +1 ⫋ lin +2 ⋅⋅⋅ ⊆ . Eventu- ally, we will reach a set which spans and hence is a basis. is possibly infinite dimensional For the general case, we can adapt the proof of the existence of a basis (Exercise 1.138), restricting to be the class of all linearly independent subsets of containing ....
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online