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A subspace of is a subset of that is also a complex vector space. The set of all

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linear combinations of a set of vectors of is a vector subspace

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called the linear span of , $

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If the ™s are linearly independent, then each vector of admits a unique expres-

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sion as a linear combination of the ™s. The set is then called a basis of the subspace $

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Given two vector subspaces and , their sum is a subspace de¬ned as the set of

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all vectors that are equal to the sum of a vector of and a vector of . The intersection

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of two subspaces is also a subspace. If the intersection of and is reduced to , then

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the sum of and is called their direct sum and is denoted by . When

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an element of and an element of . The transformation that maps into

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is a linear transformation that is idempotent, i.e., such that . It is called a projector

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onto along .

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Two important subspaces that are associated with a matrix of are its range,

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and its kernel or null space

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The range of is clearly equal to the linear span of its columns. The rank of a matrix

is equal to the dimension of the range of , i.e., to the number of linearly independent

columns. This column rank is equal to the row rank, the number of linearly independent

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rows of . A matrix in is of full rank when its rank is equal to the smallest of

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A subspace is said to be invariant under a (square) matrix whenever . In

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subspace is called the eigenspace associated with and consists of all the

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eigenvectors of associated with , in addition to the zero-vector. @

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