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.

In the figure below, ∠dec ≅ ∠dce, ∠b ≅ ∠f, and segment df is congruent to segment bd. point c is the point of intersection between segment ag and segment bd, while point e is the point of intersection between segment ag and segment df. the figure shows a polygon comprised of three triangles, abc, dec, and gfe. prove δabc ≅ δgfe.