丘维声编写的《高等代数》的课后习题详解

(

)

3

.1.P64,Ex3(1)

β

α1,α2,α3

c1α1+c2α2+c3α3=β 1c1+2c2 4c3=8 3c1+c3=3

7c2 2c3= 1 5c1 3c2+6c3= 25

c1,c2,c3

c1,c2,c3

丘维声编写的《高等代数》的课后习题详解

β=k1α1+k2α2+...+ksαs+l1α1+l2α2+...+lsαs=2)α2+...+(ks+ls)αs (k1+l1)α1+(k2+l k1+l1=k1l1=0 k2+l2=k2 l2=0

............ ...... k+l=k l=0

ssss

α1,α2,...,αs

β

α1,α2,...,αs

β

α1,α2,...,αs

α1,α2,...,αs

k1,k2,...,ks∈K

β=k1α1+k2α2+...+ksαsβ=l1α1+l2α2+...+lsαs

l1,l2,...,ls∈K

k1α1+k2α2+...+ksαs (l1α1+l2α2+...+lsαs)=0 (k1 l1)α1+(k2 l2)α2+...+(ks ls)αs=0 k1 l1 k l

22.. . k lss

α1,α2,...,αs

.3.P73,Ex8

=0l1 l=02 ..... .... l=0s

=k1=

...

k2

...

=ks

.

.

α1,...,αi 1,β,αi+1,...,αs0.

k1,...,ki 1,l,ki+1...,ks∈K

l=0.

k1α1+...+ki 1αi 1+lβ+ki+1αi+1+...+ksαs=0

l=0

k1,...,ki 1,l,ki+1...,ks

k1,...,ki 1,ki+1...,ks

α1,α2,...,αs

k1α1+...+ki 1αi 1+ki+1αi+1+...+ksαs=0

.

.

l=0.

β=b1α1+...+biαi+...+bsαsbi=0

k1α1+...+ki 1αi 1+l(b1α1+...+biαi+...+bsαs)+ki+1αi+1+...+ksαs=0 (k1+lb1)α1+...+(ki 1+lbi 1)αi 1+lbiαi+(ki+1+lbi+1)αi+1+...+(ks+

2

丘维声编写的《高等代数》的课后习题详解

lbs)αs=0

lbi=0

αi

α1,...,αi 1,αi+1,...,αs

.

..

αi(1<i≤s)

.4.P73,Ex9

α1,α2,...,αs

.

.

α1,α2,...,αs

.

α1,α2,...,αs

α1

.

α1,α2,...,αs

K(1<k≤s)

α1,...,αk 1

α1,...,αk 1,αk

(

k

α1,α2,...,αs

).αk

α1,...,αk 1

.

.

..

.

a11...

···

...

a1r

...

=0

.5.P73,Ex10

.

β1,β2,...,βr

ar1···arr

.

x1(a11α1+...+a1rαr)+...+xr(ar1α1+...+arrαr)=0 (a11x1+...+ar1xr)α1+...+(a1rx1+...+arrxr)αr=0

β1

... β

x1β1+x2β2+...+xrβr=0

=a11α1+...+a1rαr......

r

=ar1α1+...+arrαr

x1β1+x2β2+...+xrβr=0

α1,α2,...,αr

(a11x1+...+ar1xr)α1+...+(a1rx1+...+arrxr)αr=0

a11x1+···+ar1xr=0............ ax+···+ax=0

1r1

rrr

3

丘维声编写的《高等代数》的课后习题详解

x1,...,xr

a11...···

...a1r

...

T =0,

a11x+···

+a=0=

.1

..r1xr

. .... ...

..

.

ar1···arr

...

a1rx

1

+···+a=0 rrxr

x..1

.β1,β2,...,βr .

xr=0

.

.

β1,β2,...,βr

a11···

a .... .

.

.1r

..

a r1···arr

=0. β=

a

.1

.11α1+...+a .. ...1rαr

..

2+...+xrβr=0

x β

r

=ar1α1+...+arrαr

x1β1+x2β1(a11α1+...+a1rαr)+...+xr(ar1α1+...+arrαr)=0 (a11x1+...+ar1xr)α1+...+(a1rx1+...+arrxr)αr=0

α1,α2,...,αr

(

a11x1+...+ar1xr)α1+...+(a1rx1+...+arrxr)αr=0 a11x1+···+ar1xr=0. . .......... a1rx1+···+arrxr

=0

a11x1+···+ar1xr=0β1,β2,...,βr ........0

.... a1rx1+···+arrxr

=0

a .11···a1r..... ... a r1···arr .6.P 79,Ex =0.

3

α1,α2,...,αs∈Kn

rαi1,...,αir

αj1,...,αjr

.

α1,α2,...,αsr

4

.

丘维声编写的《高等代数》的课后习题详解

.

αj1,...,αjr

αt∈{α1,α2,...,αs}α1,α2,...,αs

αj1,...,αjr,αt

.

αj1,...,αjr,αt

αj1,...,αjr,αt

α1,α2,...,αs

α1,α2,...,αs

αi1,...,αir

αj1,...,αjr,αt

αi1,...,αir

.

αj1,...,αjr,αt

..

.7.P79,Ex6

α1,α2,...,αn

α1,α2,...,αn

n

.

Kn

ε1,ε2,...,εn

n

ε1,ε2,...,εn

α1,α2,...,αn

ε1,ε2,...,εn

.

α1,α2,...,αn

α1,α2,...,αn

ε1,ε2,...,εn

.

rαi1,...,αir

r

.8.P79,Ex7

α1,α2,...,αs∈Kn

α1,α2,...,αs

αi1,...,αir

.

αi1,...,αir

α1,α2,...,αs

αi1,...,αir

αi1,...,αir

r

α1,α2,...,αs

αi1,...,αir

αi1,...,αir

α1,α2,...,αs

αi1,...,αir

α1,α2,...,αs

αi1,...,αir

r

β

.9.P79,Ex8

x1α1+x2α2+...+xnαn=β

α1,α2,...,αn

β

x1α1+x2α2+...+xnαn=β

β∈Kn

α1,α2,...,αn

Ex5,Ex6

β∈Kn

α1,α2,...,αn

α1,α2,...,αn

α1,α2,...,αn

|A|=0

.10.P79,Ex9

r

αi1,...,αir

α1,α2,...,αs

αi1,...,αir

α1,α2,...,αs,β

α1,α2,...,αs,β

r

αi1,...,αir

α1,α2,...,αs,β

α1,α2,...,αs,β

αi1,...,αir

αi1,...,αir

5

丘维声编写的《高等代数》的课后习题详解

α1,α2,...,αs

α1,α2,...,αs,β

α1,α2,...,αs

β

α1,α2,...,αs

βj1,βj2,...,βjm

.11.P79,Ex10

αi1,...,αir

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