9. Let T : V → W be a linear transformation, where V and W are finite dimensional vector spaces. Show that T is onto if and only if there exists a linear transformation S : W-» V such that TS(w)= w for all w in W. (10 points)

Selection ONE of the following three topics and write a multiple (at least three) paragraph composition in English. 1. Write a composition telling about the programs, students, faculty, facilities, and extracurricular activities of an educational institution you have attended or know well. 2. Write a composition telling whether you think a housewife should receive a salary or not? 3. Write a composition telling about what you think an idea! education should be? Your composition will be given a percentile grade (with 100% as perfect score) weighing the following points: Ideas: Js the main idea visible? significant? Are subordinate ideas visible? distinct? related? sufficient? Supporting Material: Are ideas appropriately proven, explained, exemplified, documented? Is the support clear? sufficient? effective? Organization: Is the introduction appropriate? Is it sufficient to cover the paper? Is it . effective? Is the conclusion appropriate? sufficient? effective? Is the sequence of parts' visible? logical? Are transitions effective? Are paragraphs proportionate (neither too long nor too short)? Expression: Are sentences varied? effectively constructed? Is diction appropriate? accurate? Is the language used standard English? Mechanics: Are there complete sentences? Is punctuation correct? Is language usage grammatical? Is spelling correct? Is format correctly followed?

9. Let T : V → W be a linear transformation, where V and W are finite dimensional vector spaces. Show that T is onto if and only if there exists a linear transformation S : W-» V such that TS(w)= w for all w in W. (10 points)

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