Algebra 1: Math20008: Sheet 1

            G.C.Smith@bath.ac.uk

October 12, 2004

The course web site is available via http://www.bath.ac.uk/∼masgcs/

1. Let V be a vector space over F. Here is a proof that 0v = 0 for all v ∈ V. First note that 0 + 0 = 0 (Field Axiom 3). Therefore (0 + 0)v = 0v. Now by Vector Space Axiom 7 we have 0v + 0v = 0v. Now, by Vector Space Axiom 4 the vector 0v has an additive inverse −0v. It follows that

(0v + 0v) + −0v = 0v + −0v.

Now rebracket the left hand side using Vector Space Axiom 2. Therefore

0v + (0v + −0v) = 0v + −0v.

Now on both sides we have the opportunity to use the definition of an additive inverse so 0v+0 = 0. Now by Vector Space Axiom 3 we deduce that

0v = 0.

Now, using this as a model, and quoting this result if necessary, prove that if v ∈ V, then (−1)v = −v.

2. Let V be a vector space over F. Suppose that U is a subset of V. Show that U is a subspace of V if and only if the following three conditions are satisfied:

(a) U ≠ ∅.

(b) If x ∈ U and λ ∈ F, then λx ∈ U.

(c) If x, y ∈ U, then x + y ∈ U.

3. Consider the usual Cartesian description of the plane as ℝ^{2} (with perpendicular axes). This collection of ordered pairs is a vector space over ℝ in a natural way as discussed in lectures. In each case you should justify your answer.

(a) Prove that the ordered pairs corresponding a straight line through the origin form a subspace.

(b) Consider the straight line S = {(x, 1) | x ∈ ℝ}. Is this a subspace?

(c) Consider the circle C = {(x, y) | x^{2} + (y − 1)^{2} = 1}. Is this a subspace?

(d) Prove that Z = {(0, 0)} is a subspace.

(a) Prove that ∅ is not a subspace.

4. Describe as many different subspaces of ℝ^{3} as you can find.

5. Let ℝ[X] denote the set of polynomials in X which have coefficients in ℝ. This set has a natural vector space structure over ℝ. Which of the following are subspaces of ℝ[X], and why?

(a) {f | f ∈ ℝ[X], f(42) = 0}.

(b) {f | f ∈ ℝ[X], f(42) = 1}.

(c) {f | f ∈ ℝ[X], f has at most two real roots }.

(d) {f | f ∈ ℝ[X], deg f ≤ n} where n ∈ ℕ∪{0}. Note that the degree of the zero polynomial is −∞, a symbol deemed to be smaller than all integers.

(e) {f | f ∈ ℝ[X], f′′(X) − Xf′(X) + f(X) = 0} where a dash in the exponent indicates differentiation with respect to X.

(f) {f | f(X)^{2} = f(X^{2})}.

6. Suppose that V is a vector space and that v_{1}, v_{2}, …, v_{m} ∈ V. Let

U = {∑_{i=1}^{n} λ_{i}v_{i} | λ_{i} ∈ ℝ for all 1 ≤ i ≤ n} .

Show that U is a subspace of V.

7. Suppose that U, W are subspaces of V and that U ∪W is also a subspace of V. Prove that either U ⊆ W or W ⊆ U.