# prove or disprove: statement about Span

• Dec 28th 2012, 04:18 AM
Stormey
prove or disprove: statement about Span
$V$ is a linear space. $A,B\in V$ are sets, and $W$ is subspace:

$Span(A\cup B\cup W)=(Span(A)+Span(B))\cup W$

i just need help to figure out if it's true or false.

• Dec 28th 2012, 04:31 AM
HallsofIvy
Re: prove or disprove: statement about Span
Do you know what "span" means? It looks to me like a straight forward case of applying the definition.

What have you tried?
• Dec 28th 2012, 10:25 AM
Deveno
Re: prove or disprove: statement about Span
suppose V is the vector space R3, A = {(0,0,1)} B = {(0,1,0)} and W is the x-axis (vectors of the form (x,0,0)). what is the LHS and RHS then?
• Dec 29th 2012, 01:17 AM
Stormey
Re: prove or disprove: statement about Span
Hi guys, and thank you.

Denovo, that's the first example that crossed my mind, by this example the LHS and RHS are equal:
$Span\left \{ (0,1,0), (0,0,1), (x,0,0)|x\in \mathbb{R} \right \}=\left \{ (x,y,z)|x,y,z\in \mathbb{R}\right \}=\mathbb{R}^3$
is equal to:
$Span(A)=\left \{ (0,y,0)|y\in \mathbb{R}\right \}+Span(B)=\left \{ (0,0,z)|z\in \mathbb{R}\right \}\cup \left \{ (x,0,0)|x\in \mathbb{R}\right \}=\left \{ (x,y,z)|x,y,z\in \mathbb{R}\right \}=\mathbb{R}^3$

but my intuition misled me so many times, and i wasn't sure that proves it.
• Dec 29th 2012, 03:04 AM
Deveno
Re: prove or disprove: statement about Span
no, they are not equal. the point (3,4,5) is in Span(AUBUW), since:

(3,4,5) = 4(0,1,0) + 5(0,0,1) + (3,0,0), and (0,1,0) is in A, so in AUBUW, (0,0,1) is in B, so in AUBUW, and (3,0,0) is in W, so is in AUBUW.

is (3,4,5) in Span(AUB)UW? if so, it would have to be in either Span(AUB), or W, or both. is this true?
• Dec 29th 2012, 08:28 AM
Stormey
Re: prove or disprove: statement about Span

$(3,4,5)$ is neither in $(Span(A)+Span(B))$ nor in $W$, and therefore is not in $(Span(A)+Span(B))\cup W$.

thanks.
this disproves it.

this is all so confusing... (Thinking)
• Dec 29th 2012, 08:53 AM
HallsofIvy
Re: prove or disprove: statement about Span
The best way to avoid confusion in mathematics is to learnthe definitions well!
• Dec 29th 2012, 09:28 AM
Deveno
Re: prove or disprove: statement about Span
and the moral of our story is: span and union don't play well together, and should therefore sleep in separate rooms.