Summary: Math 2090 Spring 2005
Sections 4 and 7 P. Achar
Solutions to the Exercises in Section 5.2
1. The set of rational numbers (that is, numbers that can be written as fractions) is closed under addition:
adding two fractions always gives you a fraction.
It is not closed under scalar multiplication. In particular, if you multiply an irrational scalar (such as
2) by a rational number, the answer is irrational.
2. This question is either poorly worded or a trick question. Here are the two possibilities:
They may have meant "the set of all n × n upper-triangular matrices" for some n. This set is closed
under both addition and scalar multiplication.
On the other hand, if they genuinely meant all upper-triangular matrices (regardless of size), then
the addition operation isn't even always defined. For instance, you can't add a 2 × 2 upper-triangular
matrix to a 5 × 5 one. Since addition isn't always defined, closure under addition doesn't make sense.
However, this set is still closed under scalar multiplication.
3. Not closed under either addition or scalar multiplication, because the equation is nonhomogeneous.
4. Closed under both addition and scalar multiplication, because the equation is homogeneous.
5. Closed under both addition and scalar multiplication, because the equation is homogeneous.
6. (a) Yes. The zero vector in M2(R) is 0 = [ 0 0