A Matrix Product State (MPS) represents a quantum many-body wavefunction as a chain of tensors: |ψ⟩ = Σ A¹_{s₁} A²_{s₂} ... Aᴸ_{sₗ} |s₁s₂...sₗ⟩. The bond dimension χ controls how much entanglement can be captured — χ=1 gives a product state; χ=d^(L/2) is exact but exponential. The DMRG algorithm (White 1992) variationally optimizes each tensor while keeping the rest fixed, sweeping left-right until convergence. By the area law (Hastings 2007), gapped 1D systems have entanglement entropy S(ℓ) = O(1), so MPS with polynomial χ can represent them exactly. Gapless/critical systems have S~log(ℓ) — harder but tractable. Visualization shows: singular value spectra at each bond (bar heights = Schmidt values), entanglement entropy profile, and expected spin profile ⟨σz⟩.