Vector Subspaces, don't understand

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Vector Subspaces, don't understand...

Homework Statement

Which of the given subsets of the vector space, M₂₃, of all 2 X 3 matrices are subspaces.

(a) [a b c, d 0 0] where b = a + c

Homework Equations

Theorem 4.3

Let V be a vector space with operations + and * and let W be a nonempty subset of V. Then W is a subspace of V if and only if the following conditions hold

(a) u and v are any vectors in W, then u + v is in W.
(b) If c is any real number and u is any vector in W, then c * u is in W.

The Attempt at a Solution

First of all I'm not exactly sure what the space R³ exactly is and what to look for.

Is it all the positive numbers in x,y and z? I know what two properties to apply when trying to figure out if its a subspace but I still don't know exactly what to look for.

If someone could explain how to look at this problem, anything about vector spaces, or point me in the direction of a good website about them that would be greatly appreciated...i have yet to find one that I like. Thanks!

 

[Dick]

Your problem has nothing to do with R^3. If you take two such matrices M1 and M2 with entries a1, b1, c1, d1 (with b1=a1+c1) and a2, b2, c2, d2 (with b2=a2+c2) and add M1+M2 getting a third matrix M3 (so e.g. a3=a1+a2, etc), is it still true that b3=a3+c3? If so, that's your property (a).

 

[aostraff]

Yeah, what you're really trying to do is to determine is if the addition of any 2 elements in the W gives an element in W. Similarly in (b), you are trying to see if the '*' of a real number c to an element of W results in an element contained in W as well.