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Exercise:
The operations above are well defined. Moreover they give V/U the structure of a vector space over K.

Solution:
Proof well definedness. Saying that the multiplication by a scalar is well defined means the following: Suppose vv' and let alpha in K. Then alpha valpha v'. Indeed if v-v'in U then alpha v-alpha v'alpha v-v'in U. Similarly for the addition: Suppose v_v_' v_v_'. We need to show that v_+v_v_'+v_'. Indeed v_+v_-v_'+v_'v_-v_'+v_-v_' in U Longrightarrow v_+v_sim v_'+v_' hence v_+v_v_'+v_'. This proves that the operations + and are well defined. It remains to show they satisfy the axioms of a vector space. These follow from the axioms on V.
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Exercise:
The operations above are well defined. Moreover they give V/U the structure of a vector space over K.

Solution:
Proof well definedness. Saying that the multiplication by a scalar is well defined means the following: Suppose vv' and let alpha in K. Then alpha valpha v'. Indeed if v-v'in U then alpha v-alpha v'alpha v-v'in U. Similarly for the addition: Suppose v_v_' v_v_'. We need to show that v_+v_v_'+v_'. Indeed v_+v_-v_'+v_'v_-v_'+v_-v_' in U Longrightarrow v_+v_sim v_'+v_' hence v_+v_v_'+v_'. This proves that the operations + and are well defined. It remains to show they satisfy the axioms of a vector space. These follow from the axioms on V.
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eth, hs22, lineare algebra, proof, quotient space, vector space
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ENG (English)
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