Question

PROVE THAT IF {v, w} SPANS V, THEN {v+w, w} ALSO SPANS V

Answer 1

You can use the idea of lienar independence

Answer 2

can you tell me if im getting close. I have this idea that U = rv + sw, and T = r(v+w) + sw
so rv + rw + sw, so rv +(r+s)w, but i dont know how to make it look like a proof, i would really like a detailed explanation

Answer 3

\[r\vec{v} +s\vec{w} = \vec{u}\]

Answer 4

\[\alpha( \vec{v} + \vec{w} ) + \beta\vec{w} = \vec{t}\]

Answer 5

let \[v\in V\]

then \[v=av+bw\]

\[v=a(v+w-w)+bw\]

\[v=a(v+w)-aw+bw\]

\[v=a(v+w)+(-a+b)w\]

Answer 6

I think your first idea was better

Answer 7

but how to i take what i had and make it look formal and allow it to show what i know, i mean how can i make it look like a proof. How do I make my end result be proof like

Answer 8

\[\alpha\vec{v} +(\alpha + \beta)\vec{w} = \vec{t}\]

Answer 9

so i can say that because a +b is E R (i dont feel like getting all the equation things out) but u know what i mean). then its a span?

Answer 10

lol...i should have use a different letter than v

Answer 11

Let
\[t∈V]\


then
\[t=av+bw\]


\[t=a(v+w−w)+bw\]


\[t=a(v+w)−aw+bw\]


\[t=a(v+w)+(−a+b)w\]

is all you need to do

Answer 12

yes \[(\alpha + \beta)\in R\]

Answer 13

why do you t=a(v+w−w)+bw in there, why do you subtract w?

Answer 14

to get what I want

Answer 15

@Zarkon I did not get your proof

Answer 16

v=v+w-w

Answer 17

I wrote any vector t as a linear combination of v+w and w

Answer 18

mmm I got it now

Answer 19

i think im starting to understand too

Answer 20

Great!

Answer 21

wow i really do get it now, thanks man!