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**kaylakutie** Hi. I have some homework that I am doing - we're doing the chapter on Vector Spaces. I've got most of them, not that I'm really understanding Vector Spaces very well, but he assigns a ridiculous amount of problems and a few of them have me stumped (below). Help would be greatly appreciated.

4.2

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**1)** Determine whether the set, together with the indicated operations, is a vector space. If it is not, identify at least one of the ten vector space axioms that fails.

The set of all 2x2 nonsingular matrices with the standard operations.

I said true, it's a vector space. ????

**2)** True/False AND cite a reason.

a) To show that a set is not a vector space, it is sufficient to shwo that just one axiom is not satisfied.

Actually got this one. True - must satisfy all axioms by definition of a vector space.

b) The set of all first-degree polynomials with the standard operations is a vector space.

I'm guessing true, but unsure.

c) The set of all pairs of real numbers of the form (0,y), with the standard operations on R^2, is a vector space.

No clue. What do they mean by operations in R^2? Just adding and subtracting vectors? And try it for the 10 axioms?

**3)** Prove that in a given vector space V, the zero vector is unique.

Not that I could even do this anyways, but what do they mean by unique? There is only one <0,0> vector??? I don't get it.

**4** Prove that in a given vector space V, the additive inverse of a vector is unique.

Just as clueless as the last one.

4.3

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**1)** Determine is the subset of C(-infinity, infinity) is a subspace of C(-infinity, infinity) for the set of all functions such that f(0) = 1.

**2)** Prove that a nonempty set W is a subspace of a vector space V if and only if ax + by is an element of W for all scalars a and b and all vectors x and y in W.

**3)** Let A be a fixed m x n matrix. Prove that the set W = {x E R^3: Ax = [[1][2]]}

the [1,2] matrix there is a column - 2x1.

**4)** Give an example showing that the union of two subspaces of a vector space V is not necessarily a subspace of V.