MTH-102 Linear Algebra
Homework I
(Due 11/01)
1. Express the following complex numbers of the form x + iy, where x, y are real
numbers.
(a) (1 + 3i)−1
(b) (7 + πi)(π + i)
√
(c) ( 2i)(π + 3i)
2+i
(d)
2−i
2i
(e)
3−i
2. Let α and β are complex numbers. Then show that
(a) αβ = αβ
(b) α + β = α + β
3. Prove that for any two complex numbers z and w, we have:
(a) |z| − |w| ≤ |z − w|
(b) |z| − |w| ≤ |z + w|
4. Show that if z1 , z2 , . . . , zn are complex numbers, then
|z1 + z+ . . . + zn | ≤ |z1 | + |z2 | + . . . + |zn | .
5. Describe geometrically the following sets of points z that satisfy the following conditions.
(a) |z − i + 3| = 5
(b) |z − i + 3| ≤ 5
(c) |z + 2i| ≤ 1
(d) Im z ≥ 0
(e) Re z ≥ 0
6. Express the following complex numbers in polar form.
√
(a) 1 − i 2
(b) −5i
(c) −1 − i
7. Express the following complex numbers in the ordinary form x + iy.
(a) e−iπ
(b) πe−iπ/3
(c) πe−i5π/4
8. Let α be a complex number.
(a) Show that there two distinct complex numbers whose square is α.
(b) In general, show that if n > 0, there are n distinct complex numbers z such
that z n = α.
(c) Find the fourth root of i whose argument θ satisfies 0 < θ < π/2.
9. Describe all complex numbers z such that ez = w for some fixed complex number
w.
2