Worked Problems With Inner Products And Norms

A Reliable Order for Mixed Inner Product Problems

Many inner product problems ask for several quantities at once: an inner product, the lengths of the vectors, their distance, an angle, or whether they are orthogonal. The calculations are connected, but keeping their definitions distinct prevents common errors. A dependable order is to evaluate the inner product first, compute each induced norm from its own inner product with itself, and then use the appropriate formula for the quantity being asked for.

Let \(V\) be a real or complex vector space with an inner product \(\langle\mathbf{u},\mathbf{v}\rangle\), taken to be linear in the first slot and conjugate-linear in the second over \(\mathbb{C}\). In particular, \(\langle\mathbf{v},\mathbf{u}\rangle=\overline{\langle\mathbf{u},\mathbf{v}\rangle}\), and \(\langle\mathbf{u},\mathbf{u}\rangle\) is real and nonnegative, with equality to zero only for \(\mathbf{u}=\mathbf{0}\). In a real space, the inner product is symmetric. The induced norm is \(\|\mathbf{u}\|=\sqrt{\langle\mathbf{u},\mathbf{u}\rangle}\), and the distance between \(\mathbf{u}\) and \(\mathbf{v}\) is \(\|\mathbf{u}-\mathbf{v}\|\). Vectors are orthogonal when their inner product is zero.

For nonzero vectors in a real inner product space, the angle \(\theta\in[0,\pi]\) is specified by \(\cos\theta=\langle\mathbf{u},\mathbf{v}\rangle/(\|\mathbf{u}\|\|\mathbf{v}\|)\). In a complex inner product space, the angle convention is \(\cos\theta=\operatorname{Re}\langle\mathbf{u},\mathbf{v}\rangle/(\|\mathbf{u}\|\|\mathbf{v}\|)\). The real part is needed because the inner product can be complex. Neither angle formula applies if either vector is zero.

Useful Identities and Checks

Proof. Expand the inner product using linearity in the first slot and conjugate-linearity in the second: \(\langle\mathbf{u}-\mathbf{v},\mathbf{u}-\mathbf{v}\rangle=\langle\mathbf{u},\mathbf{u}\rangle-\langle\mathbf{u},\mathbf{v}\rangle-\langle\mathbf{v},\mathbf{u}\rangle+\langle\mathbf{v},\mathbf{v}\rangle\). The first and last terms are \(\|\mathbf{u}\|^2\) and \(\|\mathbf{v}\|^2\). If \(z=\langle\mathbf{u},\mathbf{v}\rangle\), conjugate symmetry gives \(\langle\mathbf{v},\mathbf{u}\rangle=\overline{z}\), so the two middle terms sum to \(-z-\overline{z}=-2\operatorname{Re}(z)\). The left side is \(\|\mathbf{u}-\mathbf{v}\|^2\), proving the formula. In a real space, \(z\) is real. \(\square\)

The Cauchy–Schwarz inequality, established for real and complex inner product spaces, states that \(|\langle\mathbf{u},\mathbf{v}\rangle|\le\|\mathbf{u}\|\|\mathbf{v}\|\). It is a bound on the magnitude of the inner product; it does not say that the inner product itself is positive. In a complex space, taking the real part also gives \(|\operatorname{Re}\langle\mathbf{u},\mathbf{v}\rangle|\le\|\mathbf{u}\|\|\mathbf{v}\|\).

Proof. In the real case, Cauchy–Schwarz gives \(|\langle\mathbf{u},\mathbf{v}\rangle|\le\|\mathbf{u}\|\|\mathbf{v}\|\). The denominator is positive because both vectors are nonzero. Dividing proves that the real cosine ratio lies between \(-1\) and \(1\). In the complex case, write \(z=\langle\mathbf{u},\mathbf{v}\rangle=a+ib\), where \(a,b\in\mathbb{R}\). Then \(|\operatorname{Re}z|=|a|\le\sqrt{a^2+b^2}=|z|\). Cauchy–Schwarz therefore gives \(|\operatorname{Re}z|\le\|\mathbf{u}\|\|\mathbf{v}\|\). Dividing by the positive denominator again puts the ratio in \([-1,1]\), as required. \(\square\)

Worked Calculations in Real Spaces

Complex Inner Products: Conjugation Matters

With the convention used here, the standard inner product on \(\mathbb{C}^n\) is \(\langle\mathbf{u},\mathbf{v}\rangle=\sum_{j=1}^{n}u_j\overline{v_j}\). The conjugate on the second vector is essential. For example, \(\langle\mathbf{u},\mathbf{u}\rangle=\sum_j|u_j|^2\) is real and nonnegative, as a squared norm must be.

Common Mistakes and a Practical Checklist

Omitting conjugation can reverse an orthogonality conclusion. For the last example, the incorrect calculation \(1(i)+i(1)=2i\) treats the complex inner product as if it were bilinear. The specified convention instead gives \(1\overline{i}+i\overline{1}=-i+i=0\). The correction is to conjugate the second vector before multiplying coordinate by coordinate.

Do not confuse the inner product with its magnitude or real part. Cauchy–Schwarz bounds \(|\langle\mathbf{u},\mathbf{v}\rangle|\), while the complex angle formula uses \(\operatorname{Re}\langle\mathbf{u},\mathbf{v}\rangle\). Orthogonality requires the entire inner product to be zero. In particular, an inner product with zero real part need not be zero: \(i\) has real part \(0\), but \(i\ne0\).

Use the stated inner product throughout. In the weighted real example, replacing \(2x_1y_1+x_2y_2\) with the usual dot product would give a different calculation. Likewise, an angle is undefined when one vector is zero, even though the inner product and distance formulas still make sense.

For a mixed problem, keep the following distinctions in view:

• Compute a norm from \(\langle\mathbf{u},\mathbf{u}\rangle\), then take the nonnegative square root.
• Use the absolute value of the inner product in a Cauchy–Schwarz bound.
• Use the real inner product for real angles and its real part for complex angles.
• Use \(\langle\mathbf{u},\mathbf{v}\rangle=0\) to test orthogonality.
• Check a distance calculation both from \(\|\mathbf{u}-\mathbf{v}\|\) and, when useful, from the norm expansion identity.