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If span{u,v} is the same as span{u,w} why does v and w are not scalar multiple of each others ?
 one year ago
 one year ago
If span{u,v} is the same as span{u,w} why does v and w are not scalar multiple of each others ?
 one year ago
 one year ago

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Jemurray3Best ResponseYou've already chosen the best response.0
Are you looking for a proof, or just a clarifying example?
 one year ago

olibdBest ResponseYou've already chosen the best response.0
More of a proof, In the answer I only have an exemple but it does not help me understading why v and w are not scalar multiple.
 one year ago

Jemurray3Best ResponseYou've already chosen the best response.0
Here's an idea. Let A be a vector space spanned by u and v. Then, any vector a in A can be written as \[ \vec{a} = c_1\vec{u} + c_2 \vec{v}\] define \[\vec{w} = \vec{u} + \vec{v} \] so \[\vec{v} = \vec{w}  \vec{u} \] then \[ \vec{a} = (c_1c_2)\vec{u} + c_2 \vec{w} = c_1^*\vec{u} + c_2 \vec{w} \] Therefore A is also spanned by u and w, where w is not a scalar multiple of v.
 one year ago

olibdBest ResponseYou've already chosen the best response.0
I'm confused a bit, I'm not sure I understand why we define w⃗ =u⃗ +v⃗ ?
 one year ago

olibdBest ResponseYou've already chosen the best response.0
and just to clarify, a⃗ is one of the Columns of A ?
 one year ago

Jemurray3Best ResponseYou've already chosen the best response.0
No. A is not a matrix, it is a vector space. a is just some vector that inhabits the space. Think about the xy plane. It is spanned by (1,0) and (0,1), correct?
 one year ago

olibdBest ResponseYou've already chosen the best response.0
yeah, (1,0) representing x and (0,1) representing y
 one year ago

Jemurray3Best ResponseYou've already chosen the best response.0
but the xy plane is also spanned by (1,0) and (1,1), and clearly (1,1) is not a multiple of (0,1)
 one year ago

olibdBest ResponseYou've already chosen the best response.0
I think i get it but I'll have to reflect on that. I have to go, thank you for your help ! :)
 one year ago
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