MATH 404 Midterm: MATH404_HERB-R_SPRING2005_0101_MID_EXAM

• School: University of Maryland
• Department: Mathematics
• Course: MATH 404
• Professor: All

Document Summary

Math 404 - exam 1 with solutions - march 7, 2005. Let v and w be vector spaces over a eld f . Assume that t is one-to-one and let {v1, v2, , vn} be a linearly independent set in v . Prove that {t (v1), t (v2), , t (vn)} is a linearly independent set in w . Solution: let c1, , cn f such that c1t (v1) + c2t (v2) + + cnt (vn) = 0. Because t is a linear transformation, cit (vi) = t (civi) for all i, and. 0 = c1t (v1) + c2t (v2) + + cnt (vn) = t (c1v1) + t (c2v2) + + t (cnvn) = t (c1v1 + c2v2 + + cnvn). But t is one-to-one and t (0) = 0, so t (c1v1 + c2v2 + + cnvn) = 0 implies that c1v1 + c2v2 + + cnvn = 0.