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https://texercises.com/exercise/bases-and-identity/
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Exercise:
Let V and W be finite dimensional spaces over K. Let ntextdimV mtextdimW. Let mathcalBB' be two bases for V and mathcalCC' two bases for W. Then textid_V_mathcalB^mathcalB'in textGL_nKtextid_Win textGL_mK and textid_V_mathcalB^mathcalB'lefttextid_V_mathcalB'^mathcalBright^- textid_W_mathcalC^mathcalC'lefttextid_W_mathcalC'^mathcalCright^-. Let T:Vlongrightarrow W be a linear map. Then: T_mathcalC'^mathcalB'textid_W_mathcalC'^mathcalC T_mathcalC^mathcalB textid_W_mathcalB^mathcalB'.

Solution:
Proof. Ttextid_Wcirc Tcirc textid_V textid_Wcirc Tcirc textid_V. &Longrightarrow T_mathcalC'^mathcalB' textid_W_mathcalC'^mathcalC Tcirc textid_V_mathcalC^mathcalB' textid_W_mathcalC'^mathcalC T_mathcalC^mathcalB textid_V_mathcalB^mathcalB'. Also we have: textid_V textid_Vcirc textid_W Longrightarrow I_n textid_V_mathcalB^mathcalBtextid_V_mathcalB^mathcalB' textid_V_mathcalB'^mathcalB and Longrightarrow I_n textid_V_mathcalB'^mathcalB'textid_V_mathcalB'^mathcalB textid_V_mathcalB^mathcalB' which results in Longrightarrow textid_V_mathcalB^mathcalB'lefttextid_V_mathcalB'^mathcalBright^- And similarly for textid_W_mathcalC^mathcalC'.
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Exercise:
Let V and W be finite dimensional spaces over K. Let ntextdimV mtextdimW. Let mathcalBB' be two bases for V and mathcalCC' two bases for W. Then textid_V_mathcalB^mathcalB'in textGL_nKtextid_Win textGL_mK and textid_V_mathcalB^mathcalB'lefttextid_V_mathcalB'^mathcalBright^- textid_W_mathcalC^mathcalC'lefttextid_W_mathcalC'^mathcalCright^-. Let T:Vlongrightarrow W be a linear map. Then: T_mathcalC'^mathcalB'textid_W_mathcalC'^mathcalC T_mathcalC^mathcalB textid_W_mathcalB^mathcalB'.

Solution:
Proof. Ttextid_Wcirc Tcirc textid_V textid_Wcirc Tcirc textid_V. &Longrightarrow T_mathcalC'^mathcalB' textid_W_mathcalC'^mathcalC Tcirc textid_V_mathcalC^mathcalB' textid_W_mathcalC'^mathcalC T_mathcalC^mathcalB textid_V_mathcalB^mathcalB'. Also we have: textid_V textid_Vcirc textid_W Longrightarrow I_n textid_V_mathcalB^mathcalBtextid_V_mathcalB^mathcalB' textid_V_mathcalB'^mathcalB and Longrightarrow I_n textid_V_mathcalB'^mathcalB'textid_V_mathcalB'^mathcalB textid_V_mathcalB^mathcalB' which results in Longrightarrow textid_V_mathcalB^mathcalB'lefttextid_V_mathcalB'^mathcalBright^- And similarly for textid_W_mathcalC^mathcalC'.
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eth, hs22, lineare algebra, proof, vector space
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