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University Math Lab: Systems & Structures FREE
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Bases & Dimension
Learning Objectives
- Determine if vectors form a basis
- Find dimension of subspaces
- Change basis
Key Concepts
Basis
Linearly independent set that spans the space
Dimension
Number of vectors in a basis
Theory
**Basis** for a vector space V: A set {v₁, …, vₙ} that is:
1. **Linearly independent** (no vector is a combination of others)
2. **Spanning** (every vector in V is a combination of basis vectors)
**Dimension:** dim(V) = number of basis vectors.
**Standard basis for ℝ³:** {e₁, e₂, e₃} = {⟨1,0,0⟩, ⟨0,1,0⟩, ⟨0,0,1⟩}
**Change of basis:** [v]_B = P⁻¹[v]_S where P is the change-of-basis matrix.
Examples
Are {(1,2), (3,4)} a basis for ℝ²?
Solution: det[[1,3],[2,4]] = 4-6 = -2 ≠ 0. Yes, linearly independent → basis.