Abstract Linear Algebra 18 | Orthonormal Basis

Orthonormal bases turn the hard parts of orthogonal projection into a fast, almost plug-and-play calculation. In a finite-dimensional subspace U of an inner product space V, choosing basis vectors that are mutually orthogonal and each have length 1 forces all cross-terms in inner products to vanish and makes the Gramian matrix collapse to the identity. That single structural change eliminates matrix inversion and turns the projection problem into a direct formula.

The key condition is written using the Kronecker delta: for basis vectors B_i and B_j in U, the inner product ⟨B_i, B_j⟩ equals δ_{ij}, meaning it is 1 when i=j and 0 when i≠j. Orthogonality is the “nice” part because it enforces right angles between distinct basis vectors, while the unit-length requirement ensures the squared length of each basis vector is exactly 1. Together, these properties define what comes later as an orthonormal system and, when the vectors also span U, an orthonormal basis.

This matters because orthogonal projection onto U typically requires solving a linear system derived from the Gramian matrix. For a general basis of U, the Gramian matrix is built from inner products of basis vectors with each other, and computing the projection involves inverting that matrix (or solving the equivalent system efficiently via row reduction). But when the basis satisfies ⟨B_i, B_j⟩=δ_{ij}, the Gramian matrix becomes the identity: ones on the diagonal and zeros off-diagonal. With that simplification, the coefficients in the projection are immediately determined by the right-hand side inner products.

Concretely, for any vector X in V, the orthogonal projection X_U onto U can be written as a linear combination of the basis vectors. With an orthonormal basis {B_1,…,B_K}, the coefficient in front of B_j is ⟨B_j, X⟩. So the projection takes the form X_U = Σ_{j=1..K} ⟨B_j, X⟩ B_j (often remembered as “basis vector times its inner product with X”). This is not just computational convenience—it also provides a clean conceptual decomposition of X into a component inside U and a perpendicular component outside U.

The transcript then formalizes the terminology. An orthogonal system requires only mutual orthogonality (⟨B_i,B_j⟩=0 for i≠j), while an orthonormal system adds the unit-length condition (⟨B_i,B_i⟩=1). If the vectors also form a basis for U, the system becomes an orthogonal basis or an orthonormal basis, respectively. The standard example is R^3 with the canonical unit vectors under the standard inner product, where the canonical basis is orthonormal and calculations are straightforward.

Finally, the discussion raises the natural next question: can any basis be transformed into an orthonormal one? The answer is yes, via a procedure promised for the next installment, setting up the move from arbitrary bases to orthonormal bases as a practical tool for inner product spaces.