$\displaystyle V$ is a linear space. $\displaystyle A,B\in V$ are sets, and $\displaystyle W$ is subspace:

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

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

thanks in advanced!

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

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

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

thanks in advanced! - Dec 28th 2012, 04:31 AMHallsofIvyRe: 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 AMDevenoRe: prove or disprove: statement about Span
suppose V is the vector space R

^{3}, 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 AMStormeyRe: 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:

$\displaystyle Span\left \{ (0,1,0), (0,0,1), (x,0,0)|x\in \mathbb{R} \right \}=\left \{ (x,y,z)|x,y,z\in \mathbb{R}\right \}=\mathbb{R}^3$

is equal to:

$\displaystyle Span(A)=\left \{ (0,y,0)|y\in \mathbb{R}\right \}+Span(B)=\left \{ (0,0,z)|z\in \mathbb{R}\right \}\cup \left \{ (x,0,0)|x\in \mathbb{R}\right \}=\left \{ (x,y,z)|x,y,z\in \mathbb{R}\right \}=\mathbb{R}^3$

but my intuition misled me so many times, and i wasn't sure that proves it. - Dec 29th 2012, 03:04 AMDevenoRe: 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 AMStormeyRe: prove or disprove: statement about Span
my bad.

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

thanks.

this disproves it.

this is all so confusing... (Thinking) - Dec 29th 2012, 08:53 AMHallsofIvyRe: prove or disprove: statement about Span
The best way to avoid confusion in mathematics is to learnthe

**definitions**well! - Dec 29th 2012, 09:28 AMDevenoRe: 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.