MATH 4377 Lecture Notes - Lecture 4: Linear Map, Differential Operator, Transpose
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Remark: sometimes, we just say t is linear. is a function from v to w. t is a linear transformation from v to. Definition: let v and w be vector spaces over f, and suppose (cid:1846: (cid:1846)(cid:4666)(cid:1876)+(cid:1877)(cid:4667)=(cid:1846)(cid:4666)(cid:1876)(cid:4667)+(cid:1846)(cid:4666)(cid:1877)(cid:4667) for all (cid:1876),(cid:1877) (cid:1848, (cid:1846)(cid:4666)(cid:1876)(cid:4667)=(cid:1846)(cid:4666)(cid:1876)(cid:4667) for all and (cid:1876) (cid:1848) Notation: (cid:1846):(cid:1848) (cid:1849) means (cid:1846) is a function from (cid:1848) to (cid:1849). (cid:1846):(cid:1848) (cid:1849): if t is a linear transformation, then t(0) = 0, t is linear if and only if t(cx + y) = ct(x) + t(y) for all. Matrix linear transformation from (cid:1844)(cid:3041) to (cid:1844)(cid:3040). Definition: suppose v and w are vector spaces over f, and (cid:1846):(cid:1848) (cid:1849) is linear. The null space of t, denoted n(t), is given by (cid:1840)(cid:4666)(cid:1846)(cid:4667)={(cid:1876) (cid:1848)| (cid:1846)(cid:4666)(cid:1876)(cid:4667)=(cid:882)}. The range of t, denoted r(t), is given by (cid:1844)(cid:4666)(cid:1846)(cid:4667)={(cid:1846)(cid:4666)(cid:1876)(cid:4667)| (cid:1876) (cid:1848)}. (cid:1846):(cid:1848) (cid:1849) is linear. Then (cid:1840)(cid:4666)(cid:1846)(cid:4667) is a subspace of v, and (cid:1844)(cid:4666)(cid:1846)(cid:4667) is a.
