Question

Construct a subset of XY plane R^2 that is:
a) closed under vector addition & subtraction, but not under scalar multiplication.
b) closed under scalar multiplication, but not vector addition & subtraction

Answer 1

OK I've the answers since it is a solved example. But can't understand it.
Ans:
a) set of all (u.v) such that u and v are ratios of p/q of integers.
b) set of all (u.v) where u=0 or v=0.
How can these vectors be closed for one and not closed for the other operation??

Answer 2

For part a), it is saying that all u's and all v's must be of the same ratio. So, taking the easiest ratio :) 1/1 = 1, all u's are a multiple of 1 and all v's are a multiple of 1. ie, (1,1),(2,-4),(0,17). You can add or subtract any of those vectors, and they will still be in the subset, that is a ratio of 1. But if you multiply them by any non-integer scalar, say 1/3, then u and v are no longer multiples of 1, and no longer in the subset.

Does that make sense?

For part b), all of those vectors would be on the x or y axis (including the origin) You can multiply any of these vectors by a scalar and they will still be on the axis 2*(1,0) = (2,0) or 3*(0,1) = (0,3), but if you add one vector from the x axis (0,2) with a vector from the y-axis (1,0), the sum is not on either axis, (1,2).

Answer 3

I think http://math.hecker.org/2011/08/12/linear-algebra-and-its-applications-exercise-2-1-1/ addresses this question.

Vectors with entries consisting only of integers, for example (1, 3) or (34, 7), are closed under addition and subtraction but if you multiply by a noninteger like 1/2 you get a new vector outside the set, for example, (1/2, 3/2) for the first vector listed..

On the other hand, points on the x and y axes, like (2.4, 0) or (0, 4.5), are closed under scalar multiplication but not under addition, for example (2.4, 0) + (0, 4.5) = (2.4, 4.5) which is not on either the x or y axis.

Answer 4

That seems like a better example, than the one given behind the Linear Algebra by Strang.