[線代] canonical quotient map

[cournot (competition)]

我想證的是3 而且是從R is onto--->T is onto

我的證法是
如果R是onto 那代表每個z都可以找到v+W s.t. R(v+W)=z
既然可以找到v+W 代表一定能找到一個v在V裡 s.t q(v)=v+W
但是我不知道是否可以這樣證
如果不行 想請教一下板友要怎麼證
謝謝

LetV and Z be vectors paces over a field F, let W be a subspace of V, and let

q:V àV/W be the canonical quotient map. Let T : V -à Z be a linear map such

that W is contained in ker(T ). Prove the following:

1. There exists a unique linear map R : V/W à Z such that T = Roq (i.e.

composition). (Hint: The equation T = Roq tells you how to define R. Check

that this definition is ‘well-defined’ and gives a linear map with the

desired property. Then show any other linear map with this property must be

R.)

3. The map R from part(1)is onto if and only if T is onto.

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