Exercise Sheet 1 Solutions#
1.#
(a)#
Take any
Their sum is
which does not satisfy
(Equivalently, the additive identity
(b)#
All axioms except distributivity over scalar addition fail:
Take
Unless
Therefore
2.#
(a)#
Zero vector:
Closure (addition): if
Closure (scalar mult.): for
All three conditions hold ⇒
(b)#
Pick
Then
and
Not closed under scalar multiplication ⇒ not a subspace.
3.#
For
The first components differ unless
Hence