(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Which of the following subsets of R^{3}? The set of all vectors of the form

a) (a, b, c), where a=c=0

b) (a, b, c), where a=-c

c) (a, b, c), where b=2a+1

2. Relevant equations

A real vector space is a set of elements V together with two operations + and * satisfying the following properties:

I. If u and v are elements of V, then u+v is in V (i.e., V is closed under the operation +).

(a) u+v=v+u, for u and v in V.

(b) u+(v+w)=(u+v)+w, for u, v, and w in V.

(c) There is an element 0 in V such that u+0=0+u, for all u in V.

(d) For each u in V, there is an element -u in V such that u+(-u)=0.

II. If u is any element of V and c is any real number, then c*u is in V (i.e., V is closed under the operation *).

(e) c*(u+v)=c*u+c*v, for all real numbers c and all u and v in V.

(f) (c+d)*u=c*u+d*u, for all real numbers c and d, and all u in V.

(g) c*(d*u)=(cd)*u, for all real numbers c and d and all u in V.

(h) 1*u=u, for all u in V.

Let V be a vector space and W a nonempty subset of V. If W is a vector space with respect to the operations in V, then W is called a subspace of V.

3. The attempt at a solution

I am having a VERY hard time grasping such abstract concepts. I think I understand the definitions of a vector space and subspace, but I don't really understand how to 'reason' with them, and put them into practice. I have seen lectures on opencourseware (MIT) with Dr. Strang and his use of geometric illustrations is much more easier follow, but my prof (and my textbook) uses only algebraic equations and I think that's where my problem is. :(

This also stands in the way of my learning of other concepts like Linear Independence, Basis, Dimension and Homogeneous Equations...

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# Linear Algebra: Vector Spaces, Subspaces, etc.

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