設(shè)V是已知平面M上所有向量的集合,對于映射f:V→V,a∈V,記a的象為f(a).若映射f:V→V滿足:對所有a、b∈V及任意實數(shù)λ,μ都有f(λa+μb)=λf(a)+μf(b),則f稱為平面M上的線性變換.現(xiàn)有下列命題:
①設(shè)f是平面M上的線性變換,a、b∈V,則f(a+b)=f(a)+f(b);
②若e是平面M上的單位向量,對a∈V,設(shè)f(a)=a+e,則f是平面M上的線性變換;
③對a∈V,設(shè)f(a)=-a,則f是平面M上的線性變換;
④設(shè)f是平面M上的線性變換,a∈V,則對任意實數(shù)k均有f(ka)=kf(a).
其中的真命題是 (寫出所有真命題的編號)
【答案】
分析:根據(jù)題意,對每一個命題進行推導(dǎo),看是否和已知條件相符.
解答:解:①:令λ=μ=1,則f(
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023212956843763981/SYS201310232129568437639005_DA/0.png)
+
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023212956843763981/SYS201310232129568437639005_DA/1.png)
)=f(
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023212956843763981/SYS201310232129568437639005_DA/2.png)
)+f(
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023212956843763981/SYS201310232129568437639005_DA/3.png)
)故①是真命題,
同理,④:令λ=k,μ=0,則f(k
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023212956843763981/SYS201310232129568437639005_DA/4.png)
)=kf(
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023212956843763981/SYS201310232129568437639005_DA/5.png)
)故④是真命題,
③:∵f(
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023212956843763981/SYS201310232129568437639005_DA/6.png)
)=-
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023212956843763981/SYS201310232129568437639005_DA/7.png)
,則有f(
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023212956843763981/SYS201310232129568437639005_DA/8.png)
)=-
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023212956843763981/SYS201310232129568437639005_DA/9.png)
,
f(λ
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023212956843763981/SYS201310232129568437639005_DA/10.png)
+μ
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023212956843763981/SYS201310232129568437639005_DA/11.png)
)=-(λ
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023212956843763981/SYS201310232129568437639005_DA/12.png)
+μ
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023212956843763981/SYS201310232129568437639005_DA/13.png)
)=λ•(-
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023212956843763981/SYS201310232129568437639005_DA/14.png)
)+μ•(-
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023212956843763981/SYS201310232129568437639005_DA/15.png)
)=λf
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023212956843763981/SYS201310232129568437639005_DA/16.png)
)+μf(
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023212956843763981/SYS201310232129568437639005_DA/17.png)
)是線性變換,
故③是真命題,
②:由f(
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023212956843763981/SYS201310232129568437639005_DA/18.png)
)=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023212956843763981/SYS201310232129568437639005_DA/19.png)
+
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023212956843763981/SYS201310232129568437639005_DA/20.png)
,則有f(
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023212956843763981/SYS201310232129568437639005_DA/21.png)
)=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023212956843763981/SYS201310232129568437639005_DA/22.png)
+
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023212956843763981/SYS201310232129568437639005_DA/23.png)
,
f(λ
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023212956843763981/SYS201310232129568437639005_DA/24.png)
+μ
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023212956843763981/SYS201310232129568437639005_DA/25.png)
)=(λ
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023212956843763981/SYS201310232129568437639005_DA/26.png)
+μ
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023212956843763981/SYS201310232129568437639005_DA/27.png)
)+
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023212956843763981/SYS201310232129568437639005_DA/28.png)
=λ•(
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023212956843763981/SYS201310232129568437639005_DA/29.png)
+
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023212956843763981/SYS201310232129568437639005_DA/30.png)
)+μ•(
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023212956843763981/SYS201310232129568437639005_DA/31.png)
+
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023212956843763981/SYS201310232129568437639005_DA/32.png)
)-
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023212956843763981/SYS201310232129568437639005_DA/33.png)
=λf(
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023212956843763981/SYS201310232129568437639005_DA/34.png)
)+μf(
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023212956843763981/SYS201310232129568437639005_DA/35.png)
)-
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023212956843763981/SYS201310232129568437639005_DA/36.png)
∵
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023212956843763981/SYS201310232129568437639005_DA/37.png)
是單位向量,
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023212956843763981/SYS201310232129568437639005_DA/38.png)
≠
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023212956843763981/SYS201310232129568437639005_DA/39.png)
,故②是假命題
故答案為①③④.
點評:本題考查了向量知識的命題判斷,注意向量的基本運算.
科目:高中數(shù)學(xué)
來源:2011-2012學(xué)年遼寧省大連一中高三(上)數(shù)學(xué)假期作業(yè)2(文科)(解析版)
題型:填空題
設(shè)V是已知平面M上所有向量的集合,對于映射f:V→V,a∈V,記a的象為f(a).若映射f:V→V滿足:對所有a、b∈V及任意實數(shù)λ,μ都有f(λa+μb)=λf(a)+μf(b),則f稱為平面M上的線性變換.現(xiàn)有下列命題:
①設(shè)f是平面M上的線性變換,a、b∈V,則f(a+b)=f(a)+f(b);
②若e是平面M上的單位向量,對a∈V,設(shè)f(a)=a+e,則f是平面M上的線性變換;
③對a∈V,設(shè)f(a)=-a,則f是平面M上的線性變換;
④設(shè)f是平面M上的線性變換,a∈V,則對任意實數(shù)k均有f(ka)=kf(a).
其中的真命題是 (寫出所有真命題的編號)
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