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Calculus and Beyond Homework Help
Linear Algebra: Null Space and Dimension
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[QUOTE="Ph21, post: 2165617, member: 179251"] [h2]Homework Statement [/h2] Prove that dim(nullA) = dim(null(AV)) (A is a m x n matrix, V is a n x n matrix and is invertible [h2]Homework Equations[/h2] AX=0 and AVX = 0 Null(AV) = span{X1,..Xd} Null(A) = span{V-1X1,.., V-1Xd} [h2]The Attempt at a Solution[/h2] so you need to prove that dim(null(AV) is a subset of nullA? Therefore d = dimA = dim(AV) and let nullA= span{X1,..Xd} and null(AV) = span{ X1,..Xd}. Null(AV) = span{X1,..Xd} Null(A) = span{V-1X1,.., V-1Xd} If a1(V-1X1 ) + …+a2(V-1Xd) = 0 So 0 = VV-1(t1X1 +… + t1Xd) 0 = t1X1 +… + t1Xd all of ti = 0 meaning it is linearly independent, and also span {V-1X1,.., V-1Xd} which is a basis of nullA AX = 0 so AVX=O then V0 = AVX which is in null(AV) This also means V0 = t1X1 +… + t1Xd which also means 0 = t1V-1X1 +…+ tdV-1Xd which spans nullA. Is this correct? Thank you, in advance :) [/QUOTE]
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Linear Algebra: Null Space and Dimension
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