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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 ?

Mathematics
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Are you looking for a proof, or just a clarifying example?
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.
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 \[ \vec{a} = (c_1-c_2)\vec{u} + c_2 \vec{w} = c_1^*\vec{u} + c_2 \vec{w} \] Therefore A is also spanned by u and w, where w is not a scalar multiple of v.

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I'm confused a bit, I'm not sure I understand why we define w⃗ =u⃗ +v⃗ ?
and just to clarify, a⃗ is one of the Columns of A ?
No. A is not a matrix, it is a vector space. a is just some vector that inhabits the space. Think about the x-y plane. It is spanned by (1,0) and (0,1), correct?
yeah, (1,0) representing x and (0,1) representing y
but the x-y plane is also spanned by (1,0) and (1,1), and clearly (1,1) is not a multiple of (0,1)
I think i get it but I'll have to reflect on that. I have to go, thank you for your help ! :)

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