# Show transcribed image text 1. Let V and W be vector spaces, and T: V right arrow W a linear transfo

Show transcribed image text 1. Let V and W be vector spaces, and T: V right arrow W a linear transformation. We say a linear transformation S: W right arrow V is a left inverse of T if ST = Iv, where Iv denotes the identity transformation on V. We say a linear transformation S : W right arrow V is a right inverse of T if TS = Iw, where Iw denotes the identity transformation on W. Finally, we say a linear transformation S: W right arrow V is an inverse of T if ST = Iv, and TS = Iw. When T has an inverse, we say T is invertible. Prove the following statements. (i) T has a left inverse if and only if T is injective, (ii) T has a right inverse if and only if T is surjective. (iii) Conclude T is invertible if and only if T is bijective (i.e. both injective and surjective).

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