Math 490 - Spring 2019

Homework 1

Due on Monday, Jan 21.

Recall the following facts about complex numbers. Every \(z \in \mathbb{C}\) can be written as \(a+bi\) where \(a, b \in \mathbb{R}\). The values \(a\) and \(b\) are called the real and imaginary parts of \(z\), and are sometimes denoted \(\operatorname{Re} z\) and \(\operatorname{Im} z\) (Notice that the imaginary part of \(a+bi\) is the real number \(b\), it does not include the \(i\)). The absolute value of \(z\) is defined to be \(|z| = \sqrt{a^2 + b^2}\). The complex conjugate of \(z\) is \(\bar{z} = a - bi\).

  1. For any complex number \(z = a + bi\) (where \(a, b \in \mathbb{R}\)), show that \(\bar{z} z = |z|^2\).

  2. Show that if \(z \in \mathbb{C}\) and \(|z|=1\), then \(\bar{z} = \frac{1}{z}\).

  3. Show that \(\operatorname{Re} z = \frac{1}{2}(z+\bar{z})\) and \(\operatorname{Im} z = \frac{1}{2i}(z-\bar{z})\).

  4. Prove that for vectors in an inner product space, \[\|x\pm y\|^2 = \|x\|^2 + \|y\|^2 \pm \operatorname{Re} \langle x, y \rangle.\]

  5. Prove the parallelogram identity for an inner product space. \[\|x+y\|^2 + \|x-y\|^2 = 2(\|x\|^2 + \|y\|^2).\]

  6. Suppose that \(v_1, \ldots, v_n\) is a basis for an inner product space \(V\).
    1. Prove that if \(\langle x, v_k \rangle = 0\) for all \(k \in \{1, \ldots, n\}\), then \(x = 0\).
    2. Prove that if \(\langle x, v_k \rangle = \langle y, v_k \rangle\) for all \(k \in \{1, \ldots, n\}\), then \(x = y\).

  7. Use the Law of Cosines to prove that \(\langle x, y \rangle = \|x\| \|y\| \cos \theta\), where \(\theta\) is the angle between the vectors \(x\) and \(y\) in the real subspace spanned by \(x\) and \(y\) (if you want, assume that \(x,y \in \mathbb{R}^2\)).