[問題] 請問一題線性代數(看似簡單卻解不出來 ><)

[commualg]

V is a finite-dimensional vector space

T:V-->V be a linear transformation

W is a T-invariant subspace of V

i.e. T(W) is contained in W

V=R(T)+W and R(T)��W=�r�

prove that N(T)=W


�ҩ�W�]�t��N(T)����

�i�O�Ϥ��V���]�t�N�ҩ����X��

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���� Author: commualg �q 203.71.160.160 �o��

[神犬~]

Post by commualg
V is a finite-dimensional vector space
T:V-->V be a linear transformation
W is a T-invariant subspace of V
i.e. T(W) is contained in W
V=R(T)+W and R(T)��W=�r�
prove that N(T)=W
�ҩ�W�]�t��N(T)����
�i�O�Ϥ��V���]�t�N�ҩ����X��

V is a finite-dimensional vector space
By dimension Theorem , dim(V) = dim[R(T)] + dim[N(T)]
Since V=R(T)+W and R(T)��W=�r
dim(V) = dim[R(T)] + dim(W)
Thus dim(W) = dim[N(T)]
Since W belong to N(T)
=> W = N(T)

Post by commualg
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Post by commualg
V is a finite-dimensional vector space
T:V-->V be a linear transformation
W is a T-invariant subspace of V
i.e. T(W) is contained in W
V=R(T)+W and R(T)��W=�r�

�ʡʡʡʡ�
�o�䦳���D, R(T) �O V �� subspace, W �]�O V �� subspace

�O���O�n�令 R(T)��W = {0} ? ^^a

�p�G�O�����D�إi�H�令 V = R(T)��W, �o�˴N�n���D�ܦh�F���F

Post by commualg
prove that N(T)=W

Pf:
(i) W C N(T)
��
Let x �` W , T(x) �` R(T)

and since W is a T-invariant subspace of V, T(x) �` W

=> T(x) �` R(T)��W = {0}

=> x �` N(T)
#

(ii) Since (1) V = R(T)+W and R(T)��W = {0}

(2) V is a finite-dimensional vector space

(3) dim(V) = dim(R(T)) + dim(W) - dim(R(T)��W)

= dim(R(T)) + dim(W) - 0

We can find dim(W) = dim(V) - dim(R(T))

By dimension Thm: dim(N(T)) = dim(V) - dim(R(T))

=> dim(W) = dim(N(T))


By (i) W C N(T) and dim(W) = dim(N(T)) & (2)
��
=> W = N(T)
�� #
(PS: �o�@�ӽb���b�������~�|����, �L�����V�q�Ŷ��|�����D

��~~~ �Pı�C��po�����o�� XD , �������� ���o^^ )

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