MATA23H3 Lecture : linear transformations

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17 Apr 2011
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If t : rn rm is a linear transformation. then: rn is the domain of t , rm is the codomain of t , if w rn then: the image of w under t is. T 1[w ] = {v rn (cid:12)(cid:12) t(v) w } The set t 1[{0 }] = {v rn (cid:12)(cid:12) t(v) = )} ( where 0 rm ) is called the kernel of t . www. notesolution. com. Theorem: let t : rn rm be a linear transformation, then: if v1, v2, v3, , vk rn and r1, r2, , rk r then. T (r1v1 + r2v2 + + rkvk) = r1t(v1) + r2t(v2) + + rkt(vk): t (0) = 0 where 0 rn and 0 rm. www. notesolution. com. If t : rn 7 rm is a linear transformation and v1, v2, , vk rn such. Theorem: that {t (v1), t(v2), , t(vk)} is a linearly independent set in rm. Then, {v1, v2, , vk} is linearly independent also.

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