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수학/선형대수학

[선형대수학] 영공간, 치역, 널리티, 랭크 (Null space, Range, Nullity, Rank)

xeskin 2020. 8. 20. 15:58
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Def_

Let V,W be vector spaces, and let T:VW be linear.

We define the null space N(T), the range R(T) of T.

N(T)={xV:Tx=0}

R(T)={TxW:xV}

Null space and range are called kernel and image respectively.

 

Thm_

Let V,W & T be an in the previous definition.

Then N(T),R(T) are subspaces of V,W, respectively.

 

 

Def_

Let V,W & T be as in the above theorem. Suppose N(T),R(T) are finite-dimensional.

Then we define the nullity, the rank of T.

nullity(T)=dim(N(T))N0

rank(T)=dim(R(T))N0

 

 

Ex_

Let V,W be vector spaces. Let I:VV be the identity map & T0:VW the zero map.

Then N(I)={0},R(I)=V,N(T0)=V,R(T0)={0}.

 

 

Thm_

Let V,W be vector spaces, and let T:VW be a linear map. If β={v1,,vn} is a basis for V, then R(T)=span(T(β))=span({T(v1),,T(vn)}).

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