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invariant subspace (Definition)

Let $T: V\rightarrow V$ be a linear transformation of a vector space $V$ A subspace $U\subset V$ is called a $T$ invariant subspace if $T(u)\in U$ for all $u\in U$

If $U$ is an invariant subspace, then the restriction of $T$ to $U$ gives a well defined linear transformation of $U$ Furthermore, suppose that $V$ is $n$ dimensional and that $v_1,\ldots, v_n$ is a basis of $V$ with the first $m$ vectors giving a basis of $U$ Then, the representing matrix of the transformation $T$ relative to this basis takes the form $$ \begin{pmatrix} A & B \\ 0 & C \end{pmatrix}$$ where $A$ is an $m\times m$ matrix representing the restriction transformation $T\big|_U:U\to U$ relative to the basis $v_1,\ldots, v_m$




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"invariant subspace" is owned by rmilson. [ full author list (2) ]
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See Also: linear transformation, invariant


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cyclic subspace (Definition) by CWoo
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Cross-references: transformation, matrix, vectors, basis, well defined, restriction, subspace, vector space, linear transformation
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This is version 6 of invariant subspace, born on 2002-02-15, modified 2007-06-09.
Object id is 1962, canonical name is InvariantSubspace.
Accessed 74138 times total.

Classification:
AMS MSC15-00 (Linear and multilinear algebra; matrix theory :: General reference works )

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Would like to add a proof by pjohara15 on 2008-01-19 10:17:33
Hello-

I am currently working on a proof of this and would like your permission to add it once I finish it.

PJO
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