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# 高等微積分...急

Let A be an n*n matrix and let c and x_* be points in R^n , Define the

affine mapping G:R^n→R^n by

G(x)=c+A(x- x_* ) for x in R^n

Show that the mapping G:R^n→R^n is one-to-one and onto if and only

if the matrix A is invertible

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### 1 Answer

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- mathmanliuLv 71 decade agoFavorite Answer
Jacobian J=det(A), 則

det(A)≠0 <=> the mapping G is 1-1 and onto

而 det(A)≠0 <=> A is invertible

註: onto的說明

For any y in R^n, is there x in R^n, such that G(x)=y.

i.e. c +A(x- x*)= y <=> x= A^(-1) (y-c) + x*

即本題(線性函數), 只要 1-1 就有onto

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