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

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 an…

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