# Factoid Details

### ‘An operator can be linear.’

[⟨det operator.n⟩ linear.a]

In Wikipedia:

A Fredholm operator is a bounded linear operator between two Banach spaces whose kernel and cokernel are finite – dimensional and whose range is closed.
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In Wikipedia:

The operator giving the sum of a convergent series is linear ’”, and it follows from the Hahn – Banach theorem that it may be extended to a summation method summing any series with bounded partial sums.
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In Wikipedia:

A particular consequence of this last observation is that the Riesz transform defines a bounded linear operator from “L” 2 (R “d”) to itself.
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In Wikipedia:

Instead, we have crude tests known as the energy conditions, which are similar to placing restrictions on the eigenvalues and eigenvectors of a linear operator.
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In Wikipedia:

The characteristic polynomial of any diagonalizable linear operator splits. [Formula 153] 1dim (E) m ===.
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In Wikipedia:

Let T be a linear operator on a finite – dimensional vector space V, and let and’ be two ordered bases for V. Suppose that Q is the change of coordinate matrix that changes’ – coordinates into – coordinates.
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In Wikipedia:

Since the discrete Fourier transform is a linear operator, it can be described by matrix multiplication.
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In Wikipedia:

By analogy with the term “eigenvector” for a vector which, when operated on by a linear operator is another vector which is a scalar multiple of itself, the term “eigenplane” can be used to describe a two – dimensional plane (a “2 – plane”), such that the operation of a linear operator on a vector in the 2 – plane always yields another vector in the same 2 – plane.
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In Wikipedia:

In fact, if a linear operator has the orthogonal matrix and this matrix is symmetric the “Symmetric operator theorem” valid in [Formula 40] (any dimension) applies saying that it has “n” orthogonal eigenvectors.
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In Wikipedia:

Let T be a linear operator on a vector space V, and let [Formula 146] be distinct eigenvalues of T.
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In Wikipedia:

Let [Formula 40] be a second degree linear operator.
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In Wikipedia:

In linear algebra, an involution is a linear operator “T” such that [Formula 2].
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In Wikipedia:

Recall that an eigenvector of a linear operator “A” is a (non – zero) vector “x” such that “Ax” = “x” for some scalar.
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In Wikipedia:

Just as you would expect from the theory of the eigenvalues of an ordinary linear operator on a four dimensional vector space, the different types of Weyl tensor (at a given event) can be determined by solving a certain quartic polynomial.
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In Wikipedia:

The operator [Formula 6] is linear.
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In Wikipedia:

The sum of the “k” largest singular values of “M” is a matrix norm, the Ky Fan “k” – norm of “M” The first of the Ky Fan norms, the Ky Fan 1 – norm is the same as the operator norm of “M” as a linear operator with respect to the Euclidean norms of “K’m” and “K ’n”.
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In the Brown Corpus:

If T is a linear operator on an arbitrary vector space and if there is a monic polynomial P such that A[fj], then parts (A) and (B) of Theorem 12 are valid for T with the proof which we gave.
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In Wikipedia:

To formalize it we need to say: let “T” be a linear operator defined on a family “F” of functions which is dense in both [Formula 1] and [Formula 2] (for example, the family of all simple functions).
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In Wikipedia:

Kernel (linear operator).
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And 259 more sentences.

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Queried on 2013-05-18.