Mathematics, 22.11.2019 23:31, siyah87

Suppose v is finite-dimensional and v1; : : : ; vm 2 v. define a linear map w v 0 ! fm by .'/ d '.v1/; : : : ; '.vm/ :
(a) prove that v1; : : : ; vm spans v if and only if is injective.
(b) prove that v1; : : : ; vm is linearly independent if and only if is surjective.

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Suppose v is finite-dimensional and v1; : : : ; vm 2 v. define a linear map w v 0 ! fm by .'/ d '.v1/; : : : ; '.vm/ : (a) prove that v1; : : : ; vm spans v if and only if is injective. (b) prove that v1; : : : ; vm is linearly independent if and only if is surjective.

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Suppose v is finite-dimensional and v1; : : : ; vm 2 v. define a linear map w v 0 ! fm by .'/ d '.v1/; : : : ; '.vm/ :
(a) prove that v1; : : : ; vm spans v if and only if is injective.
(b) prove that v1; : : : ; vm is linearly independent if and only if is surjective.