Linear Algebra Lecture 37 | Dimension Formula & Independent Subspaces

Описание: In this lecture we continue our study of independent subspaces and prove the dimension formula.

📌 Topics covered:
• Independent subspaces always intersect trivially (only at 0)
• Counterexample: trivial intersections don’t guarantee independence (3 subspaces in R²)
• Converse holds for 2 subspaces: independence ⇔ trivial intersection
• Main theorem:
dim(W1 + … + Wn) = dim(W1) + … + dim(Wn)
if and only if W1,…,Wn are independent

👉 This is the exact condition for when the inequality becomes equality.
Next lecture: we introduce the direct sum of subspaces