Defining matrix sum and scalar multiple in the usual way v

Question

1. Using the vector space axiomatic properties prove that (-1)x = -x. You may use ox = o'.

2. Prove or disprove that, in each of the following, V is a vector space:

a) V = all reals x > 0. Scalars are the reels. Define new operations as follows:

addition : x ⊕ y = xy
scalar multiplication : cx = x^c.

b) V: all continuous f such that ₀∫¹ f(x)dx = 0.

c) V: all f with f'' continuous and f'' + f' + e^xf = 0.

d) V: all x ∈ Cⁿ with ∑₁ⁿx_i = 1.

c) Defining matrix sum and scalar multiple in the usual way, V: the set or symmetric n x n it matrices.

3. Let V and W be two vector spaces over F.

Let V ⊕ W = ((v, w) | v ∈ V, w ∈ W}:

where we define (v₁, ω₁)+(v₂, ω₂) = ( v₁ + v₂, ω₁ + ω₂) and a(v₁, ω₁) = (av₁, aω₁)

for all v₁, v₂ ∈ V and ω₁, ω₂ ∈ W and a ∈ P. Prove that V ⊕ W is a vector space over F. (V ⊕ W is called the direct sum of V and W.)

4. In each of the following eases, establish whether or not the given set of vectors is linearly independent or linearly dependent. Explain.

a) 1, cost, oos2t, ..., cos nt

b) (t- 1)(t - 2)(t - 3), t(t - 2)(t -3), t(t - 1)(t -3), t(t - 1)(t - 2)

c) t^√2, t^e, t^Π, t ≥ 0

d) coshx, cosh(x - 1)

e) (1, 1,1, 1)^T, (0, 2,1, -1)^T, (2, -4, -1,5)^T

f) (1,1, 1,1)^T, (0,2,1,-1)^T, (2,-1,1,-1)^T