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Spanning Set Question

Hey guys i was wondering if you could help me out, heres the question with my working.

Let M e F . Show that if {v1, v2, . . . , vn} spans V then so does
{v1 + Mv2, v2, . . . , vn}.

I know any v ? V can be written as a linear combination of elements of {v1, v2, . . . , vn}, so i applied this and made a substution like so;

a1 v1 + a2 v2 + ... + an vn = <v1 , ... , vn>

So if replace v1 with v1 + Mv2, then i get

a1[v1 + Mv2] + a2 v2 ... + an vn. If i then expand all this i get

a1v1 + (a1 a2 M)v2 + ... + anvn

So by defn this spans V.

Is this correct a simple substution and re-arranging to get it in terms of the defn?

Thanks for your help.
Reply 1
Avatar for nuodai
nuodai
17
I'm not really convinced by your argument. I think the best way to go about doing this is to write v1v_1 as a linear combination of vectors from {v1+Mv2,v2,…,vn}\{ v_1 + Mv_2, v_2, \dots, v_n \} (there's an obvious way to do this), and then since you can get any vector from a set you already know spans VV, you must be able to span VV with the new set.
Reply 2
Avatar for DDave
DDave
OP
1
So you mean write

V1 = ( x1, x2, x3) + M(y1, y2, y3) = (x1 + My1, x2 + My2, x3 + My3) ?
Reply 3
Avatar for DFranklin
DFranklin
18
No. If you start with v1, and you add Mv2 to it to get v1+Mv2, then what multiple of v2 could you add to v1+Mv2 to get v1?
Reply 4
Avatar for nuodai
nuodai
17
Original post by DDave
So you mean write

V1 = ( x1, x2, x3) + M(y1, y2, y3) = (x1 + My1, x2 + My2, x3 + My3) ?


What are xix_i and yjy_j?

What I was saying is this:

For simplicity let S={v1,v2,…,vn}S = \{ v_1, v_2, \dots, v_n \}. We know that SS spans VV.

Let w=v1+Mv2w = v_1 + Mv_2 so that your new set is Snew={w,v2,…,vn}S_{\text{new}} = \{ w, v_2, \dots, v_n \}. How can you write v1v_1 as a linear combination of vectors in SnewS_{\text{new}}? Once you've done that, it means that any vector of SS can be written as a linear combination of vectors in SnewS_{\text{new}}. But we know that any vector in VV can be written as a linear combination of elements of SS, so this means that any vector in VV can be written as a linear combination of linear combinations of elements in SnewS_{\text{new}}... but a linear combination of linear combinations is itself a linear combination, so you're done.
Reply 5
Avatar for DDave
DDave
OP
1
Original post by DFranklin
No. If you start with v1, and you add Mv2 to it to get v1+Mv2, then what multiple of v2 could you add to v1+Mv2 to get v1?


Well if you added -Mv2 you would get v1, but i dont understand how that helps?
Reply 6
Avatar for Haddock3
Haddock3
Original post by DDave
Well if you added -Mv2 you would get v1, but i dont understand how that helps?


Well if you can write v1 as a linear combination of the vectors in Snew, and you still have v2, v3,...vn, then you know you can get all of the vectors in original set S from linear combinations of Snew.

Once you have all the vectors in S as linear combinations of Snew, you're done, since we already have that S spans V.

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