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Exercise:
Phi_mathcalB is an isomorphism. Phi_mathcalB:Vlongrightarrow textK^n Phi_mathcalBv:v_mathcalBin textK^n

Solution:
Proof. We first show that Phi_mathcalB is a linear map. Let abin K vv'in V. Write vv' as linear combination of the elements of mathcalB: va_v_+...+a_nv_n v'b_v_+...+b_nv_n with a_ia_i'in K. By definition v_mathcalBleftarrayc a_ vdots a_n arrayrightquad v'_mathcalBleftarrayc b_ vdots b_n arrayright. Now av+bv'aa_+bb_v_+...+aa_n+bb_nv_n hence Phi_mathcalBav+bv'av+bv'_mathcalB leftarrayc aa_+bb_ vdots aa_n+bb_n arrayright aleftarrayc a_ vdots a_n arrayright+bleftarrayc b_ vdots b_n arrayright a Phi_mathcalBv+b Phi_mathcalBv'. This proves Phi_mathcalB is linear. We now claim that Phi_mathcalB is an isomorphism. By previous results it's enough to show Phi_mathcalB is bijective. Indeed: If vin textKerPhi_mathcalB then v v_+...+ v_n Longrightarrow textKerPhi_mathcalB. This shows that Phi_mathcalB is injective. Surjectivity of Phi_mathcalB: Let leftarrayc a_ vdots a_n arrayright in K^n Put va_v_+...+a_nv_n. Then Phi_mathcalBvleftarrayc a_ vdots a_n arrayright. Longrightarrow textImPhi_mathcalBK^n.
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Exercise:
Phi_mathcalB is an isomorphism. Phi_mathcalB:Vlongrightarrow textK^n Phi_mathcalBv:v_mathcalBin textK^n

Solution:
Proof. We first show that Phi_mathcalB is a linear map. Let abin K vv'in V. Write vv' as linear combination of the elements of mathcalB: va_v_+...+a_nv_n v'b_v_+...+b_nv_n with a_ia_i'in K. By definition v_mathcalBleftarrayc a_ vdots a_n arrayrightquad v'_mathcalBleftarrayc b_ vdots b_n arrayright. Now av+bv'aa_+bb_v_+...+aa_n+bb_nv_n hence Phi_mathcalBav+bv'av+bv'_mathcalB leftarrayc aa_+bb_ vdots aa_n+bb_n arrayright aleftarrayc a_ vdots a_n arrayright+bleftarrayc b_ vdots b_n arrayright a Phi_mathcalBv+b Phi_mathcalBv'. This proves Phi_mathcalB is linear. We now claim that Phi_mathcalB is an isomorphism. By previous results it's enough to show Phi_mathcalB is bijective. Indeed: If vin textKerPhi_mathcalB then v v_+...+ v_n Longrightarrow textKerPhi_mathcalB. This shows that Phi_mathcalB is injective. Surjectivity of Phi_mathcalB: Let leftarrayc a_ vdots a_n arrayright in K^n Put va_v_+...+a_nv_n. Then Phi_mathcalBvleftarrayc a_ vdots a_n arrayright. Longrightarrow textImPhi_mathcalBK^n.
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eth, hs22, isomorphism, lineare algebra, proof, vector space
Difficulty
(3, default)
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0 (default)
Language
ENG (English)
Type
Proof
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