（1）A(x)??11??11??; （2）A(x)?11???11?x???01??x????01?; ?（3）A(x)?x?xT; （4）A(x)?x*. 其中x?K2?2,x*是x的伴随矩阵。
解 （1）由A(x?y)??11?(x??11??11??11??11??1???11?y)???01??????11?x???01??????11?y???0?A(x)?A(y)， 及A(kx)??11??11??1??11????11?(kx)???01?k1?????11?x???01??kA(x)， ?可见，映射是K2?2的线性变换。 （2）因为A(x?y)?(x?y)??11?1???01??x??1???01??y， ?A(x)?A(y)?x??11?1???01??y??1???01?， ?所以映射不是线性变换。
（3）因为A(x?y)?(x?y)?(x?y)T?x?xT?y?yT?A(x)?A(y)，
A(kx)?(kx)?(kx)T?k(x?xT)?kA(x)
所以映射是K2?2的线性变换。
（4）因为A(x?y)?(x?y)*,A(x)?A(y)?x*?y* 所以映射不是线性变换。
10.求第9题中线性变换在基E?10??01?0?11???00?,E12?0?,E?0???0?21???10?,E?0?22???0的矩阵。
解 对（1）中的线性变换A
A(E?11??10??11??10??11??11?11)????11????00????01??????10????01??????1?1???E11?E12?E21?E22,
1?1? ?0?1?下??1A(E12)????1?1A(E21)????1?1A(E22)????11??0??1??01??0??1??11??0??1??01??1??0??00??1??0??00??1??1??01??0???1??01??1???1??11??0???1??01??1???1??00??1??0??01??1??1??01??0???1??01??1???1??11??0???1??01???E12?E22， ?1?1???E11?E12?E21?E22， 1?1???E12?E22， 1??1?1所以，A(E11E12E21E22)?(E11E12E21E22)???1???1?1?1???1???10101110??1?. 0??1?010111?110??1?即所求矩阵为 0??1??11对（3）中的线性变换A
?1TA(E11)?E11?E11???0?0TA(E12)?E12?E12???0A(E21)?E21?ET210???0??1??00??2???0??00??0???0??11??0???0??10??0???1??00???2E11, 0?1???E12?E21， 0?1???E12?E21， 0?0???2E22， 2?1??0???0??10??0???0??00??0???1??0?0???1?0TA(E22)?E22?E22???0?2?0?所以，A(E11E12E21E22)?(E11E12E21E22)?0??0??x1??11求线性空间K3的线性变换A??x2??x??3011001100??2??00?即所求矩阵为??00???2??0011001100??0?. 0??2??1??0??0????x1?2x2???????????ε?0,ε?1,ε?0在基?x?x12??2??3?????1??0??0??1????x?x?x???????23????1下的矩阵.
?1??2??0???????解 A(ε1)?1,A(ε2)??1,A(ε3)?0
???????1??1??1????????1?所以A(ε1ε2ε3)?(ε1ε2ε3)1??1?2?110??1??0，即所求矩阵为1????11??2?110??0。 ?1???1??0??0??1??1??1?????????????12.已知ε1?0,ε2?1,ε3?0和η1?1,η2?1,η3?0是线性空间的两组基.
?????????????0??0??1??1??0??0?????????????如果线性变换A使得
??1??2??0???????A(ε1)?1,A(ε2)?1,A(ε3)??1
???????0??1???1???????求（1）A在基ε1,ε2,ε3下的矩阵； (2) A在基η1,η2,η3下的矩阵. 解 （1）因
??1??2??0???????A(ε1)?1??ε1?ε2,A(ε2)?1?2ε1?ε2?ε3,A(ε3)??1??ε2?ε3,
???????1???1??0?????????1?所以A(ε1ε2ε3)?(ε1ε2ε3)?1?0?2110???1???1，即所求矩阵为1????0?1??2110???1. ??1???1??1??1???????（2）因为η1??1??ε1?ε2?ε3,η2??1??ε1?ε2,η3??0??ε1,
?1??0??0????????1???ε1ε2ε3?1??1?110110所以?η1η2η3?1??0， ?0??1??0?0???1这时?ε1ε2ε3??1???η1η2η3?1??1??0???η1η2η3?0??1?01?1?1???1 ?0??于是A?η1η2η3??1??A?ε1ε2ε3?1??1?01?1???1???11???0???00??1 ??2??2111101???1??0??ε1ε2ε3?1????00??0??1???11????1???11101??0 ?0??2110??1???11????1???11101??0 ?0???0???η1η2η3?0??1??2???η1η2η3??1??0??111?1?2?即所求矩阵为?1??0?11?10??1。 ??2??13.设三维线性空间Vk的线性变换A在基ε1,ε2,ε3下的矩阵为
?a11?a?21?a?31a12a22a32a13??a23 ?a33??（1）求A在基ε3,ε2,ε1下的矩阵；
（2）求A在基ε1,kε2,ε3下的矩阵，其中k?K且k?0； （3）求A在基ε1?ε2,ε2,ε3下的矩阵.
解 （1）A(ε1)?a11ε1?a21ε2?a31ε3,A(ε2)?a12ε1?a22ε2?a32ε3,
A(ε3)?a13ε1?a23ε2?a33ε3由此得
?a13?A(ε3,ε2,ε1)?(ε3,ε2,ε1)a23??a?33?a13?即所求矩阵为?a23?a?33a12a22a32a21ka12a22a32a11??a21 ?a31??a11??a21. ?a31??kε2?a31ε3,A(kε2)?k(a12ε1?a22kkε2?a32ε3)
（2）A(ε1)?a11ε1??ka12ε1?a22kε2?ka32ε3
A(ε3)?a13ε1?a23kkε2?a33ε3由此得
??A(ε1,kε2,ε3)?(ε1,kε2,ε3)??????即所求矩阵为????a111ka21ka12a22ka32a111ka21ka12a22ka32a31a13??1a23? k?a33??a31a13??1a23?. k?a33??(3)A(ε1?ε2)?A(ε1)?A?ε2???a11?a12?ε1??a21?a22?ε2??a31?a32?ε3 ??a11?a12?(ε1?ε2)??a21?a22?a11?a12?ε2??a31?a32?ε3,
A(ε2)?a12ε1?a22ε2?a32ε3?a12(ε1?ε2)?(a22?a12)ε2?a32ε3, A(ε3)?a13ε1?a23ε2?a33ε3?a13(ε1?ε2)?(a23?a13)ε2?a33ε3,
a11?a12??所以有A(ε1?ε2,ε2,ε3)?(ε1?ε2,ε2,ε3)??a21?a22?a11?a12?a31?a32?a11?a12??即所求矩阵为?a21?a22?a11?a12?a31?a32?a12a22?a12a32??a23?a13.
?a33??a13a12a22?a12a32??a23?a13
?a33??a13?1??1??1??0?????????102114.求由向量组α1???,α2???,α3???,α4???生成的子空间V的一组规范正交
?0??1???1???1??????????2??3??1???1?基.
?1?1???0??2101312?110??1??10????0?1????1??01?11111?1?10??1??10????0?1????1??001002?1001???1?得V0??0?解 由?α1,α2,α3,α4?的一组基为α1,α2,