Let ? and W be finite dimensional vector spaces and let ?:??? bea linear transformation. We say a linear transformation ?:??? is aleft inverse of ? if ST=I_v, where ?_v denotes theidentity transformation on ?. We say a linear transformation ?:???is a right inverse of ? if ??=?_w, where ?_wdenotes the identity transformation on ?. Finally, we say a lineartransformation ?:??? is an inverse of ? if it isboth a left and right inverse of ? . When ? has an inverse, we say? is invertible.
Show that
(a) ? has a left inverse iff ? isinjective.
(b) If ? is a basis for ?V and ? is a basis for ?, then[?]^?_?(Transformation from basis ? to ?) has a left inverseiff its columns are linearly independent.
