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Give an example of a matrix A that has a left inverse but does not have a right inverse. (If BA = I then B is a left inverse of A.) 2. Give an example of a matrix A that has a right inverse but does not have a left inverse. (If AB = I then B is a right inverse of A.) Let V and W be vector spaces. If T 2 L(V;W) is invertible then T is called an isomorphism and V and W are isomorphic, written V ' W. The inverse of T is written T?1. Theorem. Let T : V ! W be an isomorphism, and let f~v1; ~v2; : : : ; ~vng be a basis for V . Then the system fT(~v1); T(~v2); : : : ; T(~vn)g is a basis for W. 3. Prove this theorem.

1. Give an example of a matrix A that has a left inverse but does not have a right inverse. (If BA I then B is a left inverse of A.) 2. Give an example of a matrix A that has a right inverse but does not have a left inverse. (If AB I then B is a right inverse of A.) Let V and W be vector spaces. If T L(V, W) is invertible then T is called an isomorphism and V and W are isomorphic, written V ~ W. The inverse of T is written T-1 Theorem. Let T: V -> W be an isomorphism, and let fvi, v2, .. . , 7,) be a basis for V. Then the system {T(6),T(b), . . . . Τ(%)} is a basis for W. 3. Prove this thcorem

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의 Give an example of matrix. A that has a left inverse but hot have a ight fnveise 2 2 O 223 -fo 2 BA P. is left inverse of A g-2, 61キし 2 l 2 2 o is not right inverse For AGiveradn example of mohix A that has right in verse but not a left inverse. 2 O 1s qht inverse :is right inverse of A 2 o . B is not left inverse for AAm- T: v- be an isomorphism Lek Then system. Tj. T(%)..TTvn13 rs basis Rorw 3. T is a linear map from V to W. spanning set for V then IT(5) TIL) , T(%)} ts spanning set for H since if Ve V sinte T is lineur T、V--+ ld is injective and lineor since is isomorp hism Also n is linearly indepeclent set in v リーST(販) 1s n linearly indepedent gubsefof- and hy injedivity

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