Choose a conjugation convention before measuring angles

Try

Complex numbers carry both magnitude and phase. Write z=a+bi with a,b real and i²=−1. Its conjugate is z̄=a−bi, and |z|²=z z̄=a²+b². For x=(1+i,2i), the squared length should be 2+4=6, not (1+i)²+(2i)²=−4+2i. Compute both expressions. The failed second answer explains why complex geometry must use conjugation rather than blindly copying a real dot product.

Predict

Throughout this unit ⟨x,y⟩ is linear in x, its first argument, and conjugate-linear in y, its second. The standard formula is Σxⱼ ȳⱼ. Predict the relation between ⟨ix,y⟩ and ⟨x,iy⟩: they are i⟨x,y⟩ and −i⟨x,y⟩. Some texts choose the opposite convention; either is valid, but mixing the two changes projection coefficients and adjoint formulas. State the convention beside a computation involving complex scalars.

Experiment

For x=(1,i) and y=(i,2), compute ⟨x,y⟩=−i+2i=i, while ⟨y,x⟩=−i. Their squared norms are 2 and 5, so the Cauchy–Schwarz comparison is 1≤10 after squaring. Replace y by 3ix and check that equality holds. Construct a table with x, y, inner product, squared magnitudes and the inequality gap. This makes phase-sensitive multiplication visible while keeping the inequality a statement about nonnegative real quantities.

Understand

An inner product satisfies linearity in the first argument, conjugate symmetry ⟨x,y⟩=conj(⟨y,x⟩), and positive definiteness: ⟨x,x⟩ is real and positive for nonzero x. Define ‖x‖=√⟨x,x⟩. For y≠0 choose c=⟨x,y⟩/‖y‖². Expanding the nonnegative quantity ‖x−cy‖² gives ‖x‖²−|⟨x,y⟩|²/‖y‖². This proves Cauchy–Schwarz. Equality means the residual is zero, so x is a scalar multiple of y. If y=0, both sides of the inequality are zero. The triangle inequality follows by expanding ‖x+y‖² and bounding the real part of ⟨x,y⟩ by its absolute value and then by ‖x‖‖y‖.

Apply

An inner product need not be the standard one. Positive weights 1 and 4 give ⟨x,y⟩=x₁ ȳ₁+4x₂ ȳ₂. Under this measurement, (1,0) has length one while (0,1) has length two. Check orthogonality of (2,1) and (2,−1): the weighted product is zero even though their ordinary dot product is three. Thus angle and nearest-point questions depend on the chosen geometry. In a nonorthonormal basis, a Gram matrix records this geometry; ordinary coordinate dot products do not automatically reproduce the original inner product.

For a linear functional represented as f(x)=⟨x,y⟩, multiplying y by i multiplies the functional by −i. This is a useful convention test before the later Riesz lesson. Another test is orthogonality: ⟨x,y⟩=0 implies ⟨y,x⟩=0 by conjugate symmetry, so orthogonality itself is symmetric despite the different scalar behavior of the arguments. The norm is invariant under multiplying by any complex scalar of magnitude one, since the scalar and its conjugate cancel.

Continue

The essential scalar facts are conjugate symmetry and positivity, not an arbitrary preference for where the bar is written. Next, orthogonal decompositions will supply unique projections and a representation of linear functionals. This unit remains finite-dimensional, where every subspace has an orthogonal complement spanning the whole space. Infinite-dimensional projection theorems require closed subspaces and completeness hypotheses that are outside this sequence.

Review Linear Algebra I when a prerequisite needs practice. Continue to Separate a vector into measured and invisible parts.