189-251B: Algebra II Assignment 1

Due: Wednesday, January 15

1. Show that the set V = R^{2}, with the addition and scalar multiplication defined by the rules

(x_{1}, y_{1}) + (x_{2}, y_{2}) := (x_{1} + x_{2}, y_{1} + y_{2}), λ · (x, y) := (λx, 0)

satisfies all the axioms of a vector space except the property that 1 · v = v for all v ∈ V.

2. Which of the following subsets of R^{3} are R-vector subspaces of R^{3}?

1. The set of vectors (x, y, z) with x ≥ 0.

2. The set of vectors (x, y, z) satisfying x + y = 2z.

3. The set of vectors satisfying x = 0 or y = 0.

4. The set of vectors satisfying x = 0 and y = 0.

5. The set of vectors with rational coordinates x, y, z.

3. Let V_{1} and V_{2} be two vector subspaces of an F -vector space V. Show that the union V_{1} ∪ V_{2} can only be a subspace of V if one of the spaces V_{i} (i = 1 or 2) is contained in the other.

4. Let V be the vector space of n × m matrices with entries in a field F. What is the dimension of V? Give an explicit basis for V over F.

5. Let v_{1}, v_{2}, and v_{3} be three linearly independent vectors in an R-vector space V. Show that the vectors v_{1} + v_{2}, v_{2} + v_{3}, and v_{3} + v_{1} are also linearly independent. What if the field R is replaced by the field Z_{2} in this question?

6. Let V be the R-vector space of all infinitely differentiable functions on the real line. Show that the function T : V → V defined by T (f) = f′ (where f′ denotes as usual the derivative of f) is a linear transformation from V to itself. Show that T is not injective, and compute its kernel. Show that T is surjective. (Hint: use the fundamental theorem of calculus!) Conclude that V is not finite dimensional.

7. A generalised vector space over a field F is a not necessarily commutative group V (so that the group operation is written using the multiplicative notation) equipped with a “scalar multiplication”

F × V → V denoted (λ, v) ↦ v^{[λ]},

satisfying the following axioms analogous to those of a usual vector space

M1 (vw)^{[λ]} = v^{[λ]}w^{[λ]} for all v, w ∈ V and λ ∈ F;

M2 v^{[λ_{1}+λ_{2}]} = v^{[λ_{1}]}v^{[λ_{2}]}, for all v ∈ V and λ_{1}, λ_{2} ∈ F;

M3 (v^{[λ_{1}]})^{[λ_{2}]} = v^{[λ_{1}λ_{2}]}, for all v ∈ V, and λ_{1}, λ_{2} ∈ F;

M4 v^{[1]} = v, for all v ∈ V.

Show that a generalised vector space is just an ordinary vector space: i.e., the group law on V is necessarily commutative. (Hint: Show that v^{[−1]} = v^{−1}, the latter expression being the inverse of v in the group V, and consider axiom M1 with λ = −1 ∈ F.)

8. Let X be a set, and let P(X) denote the power set of X, i.e., the set of all subsets of X. Define the sum of two sets to be

A + B := A ∪ B − (A ∩ B),

and define a scalar multiplication of Z_{2} on P(X) by the rule:

0 · A := ∅, 1 · A = A.

Show that P(X) with these operations is a vector space over Z_{2}. What is its dimension?