Problem 3 Show that additive inverses in vector spaces are unique, i.e. for any given vector v in a vector space V, there exists a unique v* € V such that v+v* = v*+v = : Ογ. Hint: take any vector v EV and suppose v₁ and v2 are both additive inverses of v. Try to show that v₁ = v2. Desired takeaways: Learning to prove uniqueness of an object by showing equality of potential two candidate objects.

Suppose that v = (v1, v2, ..., vn) and û = (u₁, U2, ..., Un) are a pair of n-dimensional vectors. Assume that each component of the vector is a real number, so 7 and u are both members of the set R¹. We will say that and u are "almost the same" when every component of is close to every component of ū. That is, v₁ is close to u₁, v2 is close to u2, etc (practically speaking, "close" means that their absolute difference is small). Assume that we are given the predefined predicate CloseTo(x, y) and the integer constant n. Use them to write a formal definition of the new predicate Almost The Same (7, u) which asserts that n dimensional vector is almost the same as ū. Tip: It is not legal to say i v to refer to a component of v, because is not a set. Instead, use vi to refer to the ith component of v. What set would i belong to in this case?

What is the size (height) of the input vector x to allow the multiplication/evaluation of Ax given [1 2 1 243 A 0 4 c 2 50 3 ?

Problem 6. Suppose that V₁, V2 and v3 are linearly independent vectors in a vector space V. Prove that the vectors W₁ = V₁ + V2, W2 = V₂ + V3 and w3 = V3 + V₁ are also linearly independent in V.