MATA23H3 Lecture Notes - Lecture 5: 32X, Linear Map
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Definition: 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) = 0} ( where 0 rm ) is called the kernel of t . 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. 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.
