MATH-M 303 Lecture Notes - Lecture 12: If And Only If, Linear Map
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M303 section 2. 3 notes- characterizations of invertible matrices. Let (cid:1827) and (cid:1828) be square matrices; if (cid:1827)(cid:1828)=, then (cid:1827) and (cid:1828) both invertible and (cid:1827)=(cid:1828) (cid:2869),(cid:1828)=(cid:1827) (cid:2869) (cid:882) (cid:884) Using the invertible matrix theorem, determine if (cid:1827)=[(cid:883) (cid:891)] is invertible. (cid:883) (cid:884) (cid:885) (cid:887) (cid:883: (cid:1827) ~ ef [(cid:2778) (cid:882) (cid:884) (cid:882) (cid:2778) (cid:886) (cid:882) (cid:882, pivot in every row and column; columns are linearly independent and span (cid:2871) Linear map (cid:1846): invertible if there exists a linear map (cid:1845): such that (cid:1846)((cid:1845)(cid:4666)(cid:4667))= and (cid:1845)((cid:1846)(cid:4666)(cid:4667))= for all . If (cid:1845) exists (if (cid:1846) invertible), it is unique and denoted (cid:1846) (cid:2869: ex. Define (cid:1846): (cid:2870) (cid:2870) by (cid:1846)((cid:4666)(cid:2869),(cid:2870)(cid:4667))=(cid:4666)(cid:2869),(cid:2869)+(cid:2870)(cid:4667) and find its inverse if it exists. We claim (cid:1845): (cid:2870) (cid:2870) given by (cid:1845)((cid:4666)(cid:2869),(cid:2870)(cid:4667))=(cid:4666)(cid:2869), (cid:2869)+(cid:2870)(cid:4667) is an inverse to (cid:1846) Standard matrix of (cid:1846) is (cid:1827)=[(cid:883) (cid:882)(cid:883) (cid:883)] which is invertible (determinant = 1) (cid:883) (cid:883)], which is standard matrix of (cid:1845)
