Does span {u1,u2} equal span {v1,v2}?

Question

u1=1,2,3
u2=1,0,1
v1= 1,0,0
v2= 1,1,0

I'm having quite a bit of trouble understanding span. Any help is greatly appreciated.

Answer 1

The two sets do not span the same subspace of R^3. One approach to this type of problem---where both sets contain the same number of vectors---is to see if one pair is in the span of the other. That is, can you write



v1 = au1 + bu2



and



v2 = cu1 + du2



for some scalars a,b,c, and d? If you can, then the two sets span the same space. (You can flip it and try to find u1 and u2 in terms of v1 and v2.)



Can (1,0,0) be found as a(1,2,3) + b(1,0,1)? If so, then



a + b = 1

2a = 0

3a + b = 0



but there is no solution because 2a = 0 implies a = 0 and the first and last equation give b = 1 and b = 0. No go!



So v1 is not in the space span{u1,u2}.



If we know anything at all it is that v1 is in span{v1,v2}. That should be obvious. Since v1 is not in span{u1, u2}, we can conclude that span{u1,u2} and span{v1, v2} are different spaces.



Just an FYI: span{u1,u2} is just the set of all linear combinations of the vectors u1, u2. That is, it is the set of all vectors



au1 + bu2



where you let a and b range over all of the real numbers.

Answer 2

the span of u1 and u2 is all possible linear combinations:



c1u1 + c2u2, for any two scalars c1 and c2.



one way to show that span{u1,u2} = span{v1,v2}



is to show that we can find c1,c2,c3,c4 with



v1 = c1u1 + c2u2

v2 = c3u1 + c4u2



(this shows that span{v1,v2} is contained in span{u1,u2})



and also d1,d2,d3,d4 with



u1 = d1v1 + d2v2

u2 = d3v1 + d4v2



(this shows span{u1,u2} is contained in span{v1,v2})



if either of these fail, then the two span sets are not the same.



lets just look at what c1u1 + c2u2 looks like:



c1u1 = (c1,2c1,3c1)

c2u2 = (c2,0,c2)



if we add these two together, we have:



(c1 + c2,c1,3c1 + c2)



can we form v1 in such a way? if we could then:



v1 = (1,0,0) = (c1 + c2,c1,3c1 + c2) or



c1 + c2 = 1

c1 = 0

3c1 + c2 = 0



the 2nd equation tells us c1 has to be 0. that makes the other two equations:



c2 = 1

c2 = 0



which is impossible.



so v1 is NOT in the span of u1 and u2, but IS in the span of v1 and v2

(since v1 = 1v1 + 0v2)



so the two spans cannot possibly be the same.

Answer 3

What happens if you has 2 vectors and one has 3. Like (u1,u2,u3) (v1v2)? how to you check then?