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Exercise:
Let v_ v_ in V we define v_ sim v_ iff v_-v_ in U. Show that sim defines an equivalence relation on V.

Solution:
Proof of refelxivity. vsim v because v-vin U. Proof of symmetry. If v_sim v_ Longrightarrow v_sim v_ because v_-v_-v_-v_in U. Proof of transitivity. If v_sim v_ v_sim v_ Longrightarrow v_sim v_ Indeed because v_-v_v_-v_+v_-v_in U. We denote the equivalence class of vin V by v. vv+Uv+u|uin U.
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Exercise:
Let v_ v_ in V we define v_ sim v_ iff v_-v_ in U. Show that sim defines an equivalence relation on V.

Solution:
Proof of refelxivity. vsim v because v-vin U. Proof of symmetry. If v_sim v_ Longrightarrow v_sim v_ because v_-v_-v_-v_in U. Proof of transitivity. If v_sim v_ v_sim v_ Longrightarrow v_sim v_ Indeed because v_-v_v_-v_+v_-v_in U. We denote the equivalence class of vin V by v. vv+Uv+u|uin U.
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eth, hs22, lineare algebra, proof, quotient space
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(3, default)
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ENG (English)
Type
Proof
Creator rk
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