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证明:如果(x<sup>2</sup>+x+1)|f<sub>1</sub>(x<sup>3</sup>)+xf<sub>2</sub>(x<sup>3</sup>),那么(x-1)|f<sub>1</sub>(x),f(x-1)|f<sub>2</sub>(x)。
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证明:σ(x<sub>1</sub>,x<sub>2</sub>)=(x<sub>2</sub>,-x<sub>1</sub>),τ(x<sub>1</sub>,x<sub>2</sub>)=(x<sub>1</sub>,-x<sub>2</sub>)是数域F<sup>2</sup>的两个线性变换,并求σ+τ,στ,τσ。
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证明拓扑空间X为紧致空间<sub></sub>当且仅当X的每一开覆盖<sub></sub>都有一个有限(可数)开覆盖<sub></sub>的加细.
证明拓扑空间X为紧致空间<sub><img src='https://img2.soutiyun.com/ask/2020-08-10/965935556629751.png' /></sub>当且仅当X的每一开覆盖<sub><img src='https://img2.soutiyun.com/ask/2020-08-10/965935582753245.png' /></sub>都有一个有限(可数)开覆盖<sub><img src='https://img2.soutiyun.com/ask/2020-08-10/965935613112982.png' /></sub>的加细.
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(两个空间的积空间不为<sub></sub>空间的例子.)(1) 证明实数的下限拓扑空间<sub></sub>为<sub></sub>空间.(2) 记<sub>⌘
(两个<img src='https://img2.soutiyun.com/ask/2020-08-12/96609533815185.png' />空间的积空间不为<sub><img src='https://img2.soutiyun.com/ask/2020-08-12/966095360498128.png' /></sub>空间的例子.)
(1) 证明实数的下限拓扑空间<sub><img src='https://img2.soutiyun.com/ask/2020-08-12/966095408760889.png' /></sub>为<sub><img src='https://img2.soutiyun.com/ask/2020-08-12/966095438421585.png' /></sub>空间.
(2) 记<sub><img src='https://img2.soutiyun.com/ask/2020-08-12/966095462143942.png' /></sub>为两实数下限拓扑空间的积空间,证明<img src='https://img2.soutiyun.com/ask/2020-08-12/966095492512679.png' />不为<img src='https://img2.soutiyun.com/ask/2020-08-12/966095505947448.png' />空间.
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设有独立随机变量序列X<sub>1</sub>,···,X<sub>n</sub>,···,其中X<sub>k</sub>(k=1,2,···)的分布律为证明:X<sub>1</sub>,···
设有独立随机变量序列X<sub>1</sub>,···,X<sub>n</sub>,···,其中X<sub>k</sub>(k=1,2,···)的分布律为
<img src='https://img2.soutiyun.com/ask/2020-11-25/975167437725226.jpg' />证明:X<sub>1</sub>,···,X<sub>n</sub>,···满足切比雪夫大数定律。
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证明拓扑空X为T<sub>1</sub>空间当且仅当对于X的每一点x单点集{x}恰为x的所有邻域的交.
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证明马尔可夫([俄MapKOB])定理:如果不独立的随机变量X<sub>1</sub>,X<sub>2</sub>,…X<sub>n</sub>.…足条件
证明马尔可夫([俄MapKOB])定理:如果不独立的随机变量X<sub>1</sub>,X<sub>2</sub>,…X<sub>n</sub>.…足条件
<img src='https://img2.soutiyun.com/ask/2020-09-16/969121442749712.png' />
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设函数f(x)在[0,1]上连续,且f(0)= f(1),证明一定存在x∈(0,)使得f(x<sub>0</sub>)= f(x<sub>0</sub>+).
设函数f(x)在[0,1]上连续,且f(0)= f(1),证明一定存在x∈(0,<img src='https://img2.soutiyun.com/ask/2020-12-20/977320815878019.png' />)使得f(x<sub>0</sub>)= f(x<sub>0</sub>+<img src='https://img2.soutiyun.com/ask/2020-12-20/977320902712985.png' />).
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对数列{x<sub>n</sub>},若x<sub>2k</sub>→a(k→∞),x<sub>2k+1</sub>→a(k→∞),证明: x<sub>n</sub>→a(n→∞)
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设V<sub>1</sub>,V<sub>2</sub>为欧几里得空间V的两个子空间,x,y∈V.线性流形L<sub>1</sub>=x+V<sub>1</sub>,L<sub>2</sub>=y+V<sub>2</sub>之间的距离定义为
<img src='https://img2.soutiyun.com/ask/2020-08-03/96531758721777.png' />
证明:d(L<sub>1</sub>,L<sub>2</sub>)=d(x-y,V<sub>1</sub>+V<sub>2</sub>).
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证明:如果f<sub>1</sub>(x),f<sub>2</sub>(x),...,f<sub>s-1</sub>(x)的最大公因式存在,那么f<sub>1</sub>(x),f<sub>2</sub>(x),...
证明:如果f<sub>1</sub>(x),f<sub>2</sub>(x),...,f<sub>s-1</sub>(x)的最大公因式存在,那么f<sub>1</sub>(x),f<sub>2</sub>(x),...,f<sub>s-1</sub>(x),f<sub>s</sub>(x)的最大公因式也存在,且当f<sub>1</sub>(x),f<sub>2</sub>(x),...,f<sub>s</sub>(x)全不为零时有
<img src='https://img2.soutiyun.com/ask/2021-01-05/978706698167307.jpg' />
再利用上式证明,存在多项式u<sub>1</sub>(x),u<sub>2</sub>(x),...,u<sub>s</sub>(x),使
<img src='https://img2.soutiyun.com/ask/2021-01-05/978706734798402.jpg' />
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如果曲线y=f(x)上的任一条弦都高于它所限的弧,证明不等式对于所有的x<sub>1</sub>,x<sub>2</sub>(x<sub>1</sub>≠x<sub>
如果曲线y=f(x)上的任一条弦都高于它所限的弧,证明不等式<img src='https://img2.soutiyun.com/ask/2021-01-20/980004687191136.png' />对于所有的x<sub>1</sub>,x<sub>2</sub>(x<sub>1</sub>≠x<sub>2</sub>)成立(凡具有上述特性的的数叫做凸函数)
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证明:若函数f(x,y)在R(a<sub>1</sub>≤x≤b<sub>1</sub>,a<sub>2</sub>≤y≤b<sub>2</sub>)连续,
证明:若函数f(x,y)在R(a<sub>1</sub>≤x≤b<sub>1</sub>,a<sub>2</sub>≤y≤b<sub>2</sub>)连续,
<img src='https://img2.soutiyun.com/ask/2020-11-14/974189428787492.png' />
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设{x<sub>n</sub>}是内积空间X中点列,若||x<sub>n</sub>||→||x||(n→∞),且对→切y∈X有证明
设{x<sub>n</sub>}是内积空间X中点列,若||x<sub>n</sub>||→||x||(n→∞),且对→切y∈X有<img src='https://img2.soutiyun.com/ask/2020-08-13/966183406421002.png' /><img src='https://img2.soutiyun.com/ask/2020-08-13/966183418873715.png' />证明<img src='https://img2.soutiyun.com/ask/2020-08-13/966183436811741.png' /><img src='https://img2.soutiyun.com/shangxueba/ask/50580001-50583000/50580716/spacer.gif' />
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设a<sub>1</sub>,a<sub>2</sub>,…,a<sub>n</sub>为互不相同的效,F(x)=(x-a<sub>1</sub>)(x-a<sub>2</sub>)…(x-a<sub>n</sub>)。证明:任何多
设a<sub>1</sub>,a<sub>2</sub>,…,a<sub>n</sub>为互不相同的效,F(x)=(x-a<sub>1</sub>)(x-a<sub>2</sub>)…(x-a<sub>n</sub>)。证明:任何多项式f(x)用F(x)除所得的余式为
<img src='https://img2.soutiyun.com/ask/2020-07-30/964972727738352.png' />
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设f<sub>1</sub>(x)...,f<sub>m</sub>(x),g<sub>1</sub>(x),...,g<sub>n</sub>(x)都是多项式,且(f<sub>i</sub>(x)g<sub>j</sub>(x))=1(i=1,...,m;j=1,…,n),证明:(f<sub>1</sub>(x)f<sub>2</sub>(x)…fm(x),g<sub>1</sub>(x)g<s
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证明:函数f(x)在区间I单调,且x<sub>1</sub><x<sub>2</sub><x<sub>3</sub>,有[f(x<sub>3</sub>)-f(x<sub>2</sub>)][f(x<sub>2</sub>)-f(x<sub>1
证明:函数f(x)在区间I单调,<img src='https://img2.soutiyun.com/ask/2020-11-11/973942720007376.png' />且x<sub>1</sub><x<sub>2</sub><x<sub>3</sub>,有
[f(x<sub>3</sub>)-f(x<sub>2</sub>)][f(x<sub>2</sub>)-f(x<sub>1</sub>)]≥0.
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设X<sub>1</sub>,…,X<sub>n</sub>是来自泊松分布P(λ)的样本,证明:λ的近似1-α置信区间为
设X<sub>1</sub>,…,X<sub>n</sub>是来自泊松分布P(λ)的样本,证明:λ的近似1-α置信区间为
<img src='https://img2.soutiyun.com/ask/2020-09-17/969215722059981.jpg' />
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设f<sub>1</sub>(x), f<sub>2</sub>(x); g<sub>1</sub>(x), g<sub>2</sub>(x)都是数域K上的多项式,共中f<sub>1</sub>(x)≠0证明:如果g<sub>1</sub>(x)g<sub>2</sub>(x) | f<sub>1</sub>(x)f<sub>2</sub>(x), f<sub>1</sub>(x)|g<sub>1
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证明:对于一个马氏链...X<sub>0</sub>,X<sub>n-1</sub>,X<sub>n</sub>...有H(X<sub>0</sub>|X<sub>n</sub>)≥H(X<sub>0</sub>|X<sub>n-1</sub>)
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设拓扑空间<sub></sub>为T<sub>1</sub>空间,∞为任一不属于X的元素.令验证<sub></sub>为X*的拓扑,并且拓扑空间<sub></sub>为T
设拓扑空间<sub><img src='https://img2.soutiyun.com/ask/2020-08-10/965937797043895.png' /></sub>为T<sub>1</sub>空间,∞为任一不属于X的元素.令
<img src='https://img2.soutiyun.com/ask/2020-08-10/965937843858573.png' />
验证<sub><img src='https://img2.soutiyun.com/ask/2020-08-10/965937876004412.png' /></sub>为X*的拓扑,并且拓扑空间<sub><img src='https://img2.soutiyun.com/ask/2020-08-10/96593793366871.png' /></sub>为T<sub>0</sub>而非T<sub>1</sub>空间.
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设实二次型,证明:f(x<sub>1</sub>,x<sub>2</sub>,...,x<sub>n</sub>)的秩等于矩阵。的秩。
设实二次型<img src='https://img2.soutiyun.com/ask/2021-01-06/978797001811459.jpg' /><img src='https://img2.soutiyun.com/ask/2021-01-06/978797017600362.jpg' />,证明:f(x<sub>1</sub>,x<sub>2</sub>,...,x<sub>n</sub>)的秩等于矩阵。
<img src='https://img2.soutiyun.com/ask/2021-01-06/978797059757774.jpg' />
的秩。
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设f(x),g<sub>1</sub>(x),g<sub>2</sub>(x)∈C[x],证明:R(f,g<sub>1</sub>g<sub>2</sub>)=R(f,g<sub>1</sub>)R(f,g<sub>2</sub>)。
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设总体二阶矩存在,X<sub>1</sub>,…,X<sub>n</sub>是样本,证明的相关系数为-(n-1)<sup>-1</sup>.
设总体二阶矩存在,X<sub>1</sub>,…,X<sub>n</sub>是样本,证明<img src='https://img2.soutiyun.com/ask/2020-08-04/965388722721163.png' />的相关系数为-(n-1)<sup>-1</sup>.