Page 33 - Linear Algebra Done Right

Exercises
Exercises
1. Suppose a and b are real numbers, not both 0. Find real numbers 19
c and d such that
1/(a + bi) = c + di.
2. Show that √
−1 + 3i
2
is a cube root of 1 (meaning that its cube equals 1).
3. Prove that −(−v) = v for every v ∈ V.

4. Prove that if a ∈ F, v ∈ V, and av = 0, then a = 0or v = 0.

3
5. For each of the following subsets of F , determine whether it is
3
a subspace of F :
3
(a) {(x 1 ,x 2 ,x 3 ) ∈ F : x 1 + 2x 2 + 3x 3 = 0};
3
(b) {(x 1 ,x 2 ,x 3 ) ∈ F : x 1 + 2x 2 + 3x 3 = 4};
3
(c) {(x 1 ,x 2 ,x 3 ) ∈ F : x 1 x 2 x 3 = 0};
3
(d) {(x 1 ,x 2 ,x 3 ) ∈ F : x 1 = 5x 3 }.

2
6. Give an example of a nonempty subset U of R such that U is
closed under addition and under taking additive inverses (mean-
2
ing −u ∈ U whenever u ∈ U), but U is not a subspace of R .
2
7. Give an example of a nonempty subset U of R such that U is
2
closed under scalar multiplication, but U is not a subspace of R .
8. Prove that the intersection of any collection of subspaces of V is
a subspace of V.

9. Prove that the union of two subspaces of V is a subspace of V if
and only if one of the subspaces is contained in the other.

10. Suppose that U is a subspace of V. What is U + U?

11. Is the operation of addition on the subspaces of V commutative?
Associative? (In other words, if U 1 ,U 2 ,U 3 are subspaces of V,is
U 1 + U 2 = U 2 + U 1 ?Is (U 1 + U 2 ) + U 3 = U 1 + (U 2 + U 3 )?)