Finite Dimensional Spaces

Chapter 6 • Section 6-4

"Extending, reducing, and combining spaces. 🏗️"

📜 Fundamental Theorems

Extending Independent Sets

Any linearly independent set in a finite dimensional space can be extended to a basis.

Reducing Spanning Sets

Any spanning set in a finite dimensional space can be reduced to a basis.

🛠️ Building a Basis

Algorithm 1: Extending

Start with an independent set $S$.

  1. If $\span(S) = V$, stop. $S$ is a basis.
  2. Else, pick $\vec{v} \notin \span(S)$.
  3. Add $\vec{v}$ to $S$. Repeat.

Algorithm 2: Reducing

Start with a spanning set $S$.

  1. If $S$ is independent, stop. $S$ is a basis.
  2. Else, find $\vec{v}$ that is a combo of others.
  3. Remove $\vec{v}$ from $S$. Repeat.

The "n Vectors" Theorem

Let $\dim(V) = n$. If a set $S$ has exactly $n$ vectors, then:

$$ S \text{ is independent} \iff S \text{ spans } V \iff S \text{ is a basis} $$

This is a huge shortcut! You only need to check ONE condition if the count is right.

Sums and Intersections

If $U$ and $W$ are subspaces of $V$, then:

  • $U \cap W$ is a subspace.
  • $U + W = \{ \vec{u} + \vec{w} \mid \vec{u} \in U, \vec{w} \in W \}$ is a subspace.

The Dimension Formula

$$ \dim(U+W) = \dim(U) + \dim(W) - \dim(U \cap W) $$

This is like the inclusion-exclusion principle for vector spaces!

Dimension Formula Calculator

Calculate the dimension of the sum $U+W$ given the dimensions of $U$, $W$, and their intersection.

🧠 Knowledge Check Win $15

If $U$ and $W$ are planes through the origin in $\mathbb{R}^3$ ($dim=2$), and $U \neq W$, what is $\dim(U \cap W)$?

Hint: Two distinct planes in 3D intersect in a line.