Proofs; Can someone explain this to me

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[vivid mesa]

def:

I'd first like to understand by what the definition means by "written in only one way as a sum"

[steep ivy]

means if X = U + V is a direct sum, then x = u+v = u'+v', where u,u' in U and v,v' in V, implies that u=u' and v=v'

a sum of two subspaces is a direct sum if and only if their intersection contains only the zero

[steep ivy]

as per definition, the same vector is written in two different ways, hence the sum cannot be a direct sum

@vivid mesa

[vivid mesa]

thanks a lot

[hoary lodge]

@vivid mesa has given 1 rep to @steep ivy

[vivid mesa]

would you mind explaining why the third vector in the example is (0,0,0) instead of (0,y,y) like it initially is?

[steep ivy]

in both cases, the third term on the right hand side is in the form (0,y,y)

in the first case y=-1, in the second case y=0

@vivid mesa is this what you meant?

[vivid mesa]

yeah

or in

why does the vector become the zero vector

the third one that is

[steep ivy]

because for that choice of vectors, the equality holds

[vivid mesa]

+close