Question

1.Show that (u; v) = u1v1 + 2u2v2 + 3u3v3 is an inner product on R3 for u = (u1; u2; u3) and v = (v1; v2; v3).

Answer 1

An inner product must satisfy three properties.
\[{\bf{\text{Conjugate symmetry}}}\\
\text{For vectors }u,v,\text{ the inner product }\langle u,v\rangle=\overline{\langle v,u\rangle}.\]
In the case of \(\mathbb{R}^n\), you have \(\langle u,v\rangle=\langle v,u\rangle\).

\[{\bf{\text{Linearity}}}\\
\text{For vectors }u,v,w\text{ and }a,b\in\mathbb{R},\text{ you have }\langle au+bv,w\rangle=a\langle u,w\rangle+b\langle v,w\rangle.\]
Notice that linearity generally only applies to the first argument (vector). In the case of \(\mathbb{R}^n\), it also applies to the second vector as a result of conjugate symmetry.

\[{\bf{\text{Positive-definiteness}}}\\
\text{For any vector }u,\text{ the inner product with itself is always non-negative, i.e.}\\
\langle u,u\rangle\ge0\text{ and }\langle u,u\rangle=0\text{ implies }u=0.\]