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수학/선형대수학

[선형대수학] 행렬 연산의 성질 (Properties of Matrix Operation)

xeskin 2020. 8. 26. 17:19
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Thm_

Let AMm×n,B,CMn×p,D,EMq×m.

Then

(a) A(B+C)=AB+AC,(D+E)A=DA+EA

(b) a(AB)=(aA)B=A(aB)  (aF)

(c) ImA=A=AIn

(d) If V is an n-dimensional vector space with an ordered basis, then [IV]β=In.

 

Proofof(c)_

(ImA)ij=mk=1(Im)ikAkj=mk=1δikAkj=Aij

 

Coro_

Let AMm×n,B1,,BkMn×p,C1,,CkMq×m,a1,,akF.

Then

A(ki=1aiBi)=ki=1aiABi=(ki=1aiCi)A=ki=1aiCiA

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