Thread: Subspace Test

Use the subspace test to show whether or not W is a subspace of V:

1) V = R^3 and W = {7t, -3t, t x sqrt(5) | t E R}
2) V = R^3 and W = {x,y,z E V | xy + yz + xz = 0}
3) V = P2(R) (real polynomial functions of degree at most 2) and W = {f E V | f(1) = f(-1)}

Hi,
These are the problems presented. Now I have a clear idea of what the subspace test is and the conditions that must be met but I don't know what to do. Now this is a big step up from what I have been doing and the new material has somewhat phased me so please someone help me with a way into this. For a start what is R^3?

R^3 is the set of all triples (or 3-tuples) of real numbers: RxRxR. an element of R^3 is usually written like so: (a,b,c). it is made into a vector space by the addition:

(x1,x2,x3) + (y1,y2,y3) = (x1+y1,x2+y2,x3+y3), or coordinate-wise addition, and the scalar (scaling) multiplication:

a(x1,x2,x3) = (ax1,ax2,ax3).

Originally Posted by Shizaru

For a start what is R^3?

R^1 can be thought of as the "Real line" - an infinitely long line, with a point (the origin) in the 'middle'. Numbers are represented by their distance from this point.

R^2 Can be visualied as the familar cartesian plane, with x and y coordinates. Any point on this plane can be represented by two coordinates.

R^ 3 Is the x and y plane with another axis, the z axis, extending out at right angles to both the x and y axes. You now require 3 coordinates to identify a point.

If you draw an x and y axes on a piece of paper you can visualise the z axis coming out of the paper.

Originally Posted by Deveno

R^3 is the set of all triples (or 3-tuples) of real numbers: RxRxR. an element of R^3 is usually written like so: (a,b,c). it is made into a vector space by the addition:

(x1,x2,x3) + (y1,y2,y3) = (x1+y1,x2+y2,x3+y3), or coordinate-wise addition, and the scalar (scaling) multiplication:

a(x1,x2,x3) = (ax1,ax2,ax3).

Thanks,

So what am I looking to show? I know what makes it a subspace but how would I go about showing it?

You'd need to show that W obeys the same 'rules' of addition and scalar multiplication, right? Is there a form I need to rearrange it to or something?

We are only given notes which outline the basic concepts/theorems but no indication of how to put them into practice with real problems. And we can't ask the profs for help :S

well, one could verify every vector space axiom for W directly, but most of the properties that the operations (vector addition and scalar multiplication) that W has, will be exactly the same as V, since they are, in fact, the same operations. there are 3 properties that don't automatically follow from the axioms holding for V:

1) given u in W and v in W, there is no guarantee that u+v is in W (we know it will be in V). but if W is to be a vector space in its own right, the vector sum for V, has to "induce" a vector sum on W. in other words, we need to verify that we have closure of vector addition on W, so if u,v are in W, u+v has to be as well, for W to be a subspace.

2) given u in W, and a any scalar (which for your 3 vector spaces, is the field of real numbers), we have to have au in W, whenever u is in W. predictably, this is called "closure of scalar multiplication."

3) finally, W has to have something in it, as the empty set is not a vector space (it has no identity element, for one). the most convenient way to prove this is to show that the 0-vector of V lies within W (if it doesn't W has no chance of being a vector (sub-)space anyways).