1.5 Problems
Problem 1. Not all vector spaces are infinite. There are some that only contain a finite number of vectors, as we shall see next in this problem. Define the set
where the operations +,⋅ are defined by the rules
| 0 + 0 | = 0 | ||
| 0 + 1 | = 1 | ||
| 1 + 0 | = 1 | ||
| 1 + 1 | = 0 |
and
| 0 ⋅ 0 | = 0 | ||
| 0 ⋅ 1 | = 0 | ||
| 1 ⋅ 0 | = 0 | ||
| 1 ⋅ 1 | = 1. |
This is called binary (or modulo-2) arithmetic.
(a) Show that (ℤ2,ℤ2,+,⋅) is a vector space.
(b) Show that(ℤ2n,ℤ2,+,⋅) is also a vector space, where ℤ2n is the n-fold Cartesian product
and the addition and scalar multiplication are defined elementwise:
| x+ y | = (x1 + y1,…,xn + yn), x,y ∈ℤ2n, | ||
| cx | = (cx1,…,cxn), c ∈ℤ2. |
Problem 2. Are the following vector sets linearly independent?
(a) S1 = {(1,0,0),(1,1,0),(1,1,1)}⊆ℝ3
(b) S2 = {(1,1,1),(1,2,4),(1,3,9)}⊆ℝ3
(c) S3 = {(1,1,1),(1,1...
