MATH 3A Study Guide - Midterm Guide: Row Echelon Form, Surjective Function, Linear Independence
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MATH 3A Full Course Notes
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During the exam, also give the compu- tations. Hint: you can usually check if your answers are correct very quickly! Exercise 1 (a) the most common answer would be x4[ 1, 0, 1, 1, 0]t +x5[ 1/2, 1/2, 1/2, 0, 1]t + [ 1/2, 1/2, 3/2, 0, 0]t (obtained from reduced row echelon form). Another possi- ble answer is x4[1, 0, 1, 1, 0]t + x5[ 1, 1, 1, 0, 2]t + [0, 0, 1, 0, 1]t (this represen- tation is not unique). (b) from (a) it follows that the map is not one-to-one. The reduced row echelon form shows that map is not onto (there is a zero row). A = (b) the reduced row echelon form is: No zero rows: onto, no free variables: one-to-one. (c) (d) solve linear equations to nd s(e1), s(e2) and s(e3). If m > n, then m vectors in rn are always linearly dependent (theorem in book). (b) false.
