Learned by Reading
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.
(Parse)
(S1 (S (NP (DT A) (NNP Fredholm) (NN operator))
(VP (AUX is)
(NP (NP (DT a) (JJ bounded) (JJ linear) (NN operator))
(PP (IN between)
(NP (NP (CD two) (NNP Banach) (NNS spaces))
(SBAR (SBAR (WHNP (WP$ whose) (NX (NX (NN kernel)) (CC and) (NX (NN cokernel))))
(S (VP (AUX are)
(ADJP (ADJP (JJ finite)) (: -)
(ADJP (JJ dimensional)))
(CC and)
(SBAR (WHNP (WP$ whose) (NN range))
(S (VP (AUX is)
(ADJP (JJ closed))))
(. .)))
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.
(Parse)
(S1 (S (S (NP (NP (DT The) (NN operator))
(VP (VBG giving)
(NP (NP (DT the) (NN sum))
(PP (IN of)
(NP (DT a) (JJ convergent) (NN series)))
(VP (AUX is)
(ADJP (JJ linear) ('' ') ('' ''))))
(, ,) (CC and)
(S (NP (PRP it))
(VP (VBZ follows)
(PP (IN from)
(NP (DT the)
(ADJP (NNP Hahn) (: -) (NNP Banach)) (NN theorem)))
(SBAR (IN that)
(S (NP (PRP it))
(VP (MD may)
(VP (AUX be)
(VP (VBN extended)
(PP (TO to)
(NP (NP (DT a) (NN summation) (NN method))
(VP (VBG summing)
(NP (NP (DT any) (NN series))
(PP (IN with)
(NP (JJ bounded) (JJ partial) (NNS sums)))))
(. .)))
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.
(Parse)
(S1 (S (NP (NP (DT A) (JJ particular) (NN consequence))
(PP (IN of)
(NP (DT this) (JJ last) (NN observation))))
(VP (AUX is)
(SBAR (IN that)
(S (NP (DT the) (NNP Riesz))
(VP (VB transform)
(SBAR (S (VP (VBZ defines)
(NP (DT a) (JJ bounded) (JJ linear) (NN operator))
(PP (IN from) ('' '')
(NP (NNP L)))
('' '')
(NP (NP (CD 2)) (PRN (-LRB- -LRB-) (NN R) ('' '')
(NP (NNP d)) ('' '') (-RRB- -RRB-)))
(PP (TO to)
(NP (PRP itself))))
(. .)))
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.
(Parse)
(S1 (S (ADVP (RB Instead)) (, ,)
(NP (PRP we))
(VP (AUX have)
(NP (NP (JJ crude) (NNS tests))
(VP (VBN known)
(PP (IN as)
(NP (NP (DT the) (NN energy) (NNS conditions)) (, ,)
(SBAR (WHNP (WDT which))
(S (VP (AUX are)
(ADJP (JJ similar)
(PP (TO to)
(S (VP (VBG placing)
(NP (NP (NNS restrictions))
(PP (IN on)
(NP (NP (DT the) (NNS eigenvalues) (CC and) (NNS eigenvectors))
(PP (IN of)
(NP (DT a) (JJ linear) (NN operator)))
(. .)))
In Wikipedia:
The characteristic polynomial of any diagonalizable linear operator splits. [Formula 153] 1dim (E) m ===.
(Parse)
(S1 (S (NP (NP (DT The) (JJ characteristic) (NN polynomial))
(PP (IN of)
(NP (S (NP (DT any) (JJ diagonalizable) (JJ linear) (NN operator))
(VP (VBZ splits)) (. .)) (JJ -LSB-) (NN Formula))))
(NP (NP (CD 153) (JJ -RSB-) (NN 1dim)) (PRN (-LRB- -LRB-)
(NP (NNP E) (NNP )) (-RRB- -RRB-)))
(VP (VBD m)
(NP (CD ===)))
(. .)))
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.
(Parse)
(S1 (S (VP (VB Let)
(S (NP (NNP T))
(VP (VP (AUX be)
(NP (DT a) (JJ linear) (NN operator))
(PP (IN on)
(NP (NP (DT a) (NN finite)) (: -)
(NP (JJ dimensional) (NN vector) (NN space) (NN V)))))
(, ,) (CC and)
(VP (VB let)
(S (NP (NNP ) (NNP ) (CC and) (NNP ) (NNP ) (POS '))
(VP (AUX be)
(NP (NP (CD two) (VBN ordered) (NNS bases))
(PP (IN for)
(NP (NNP V.)))))
(VP (VB Suppose)
(SBAR (IN that)
(S (NP (NNP Q))
(VP (AUX is)
(NP (NP (DT the) (NN change))
(PP (IN of)
(S (VP (VB coordinate)
(NP (NNP matrix)))))
(SBAR (WHNP (WDT that))
(S (NP (NP (NNS changes))
(NP (NNP ) (NNP ) (POS ')))
(: -)
(S (VP (VBZ coordinates)
(PP (IN into)
(NP (NNP ) (NNP )))))
(: -)
(VP (VBZ coordinates)))))
(. .)))
In Wikipedia:
Since the discrete Fourier transform is a linear operator, it can be described by matrix multiplication.
(Parse)
(S1 (S (SBAR (IN Since)
(S (NP (DT the) (JJ discrete) (NNP Fourier))
(VP (VB transform))))
(VP (AUX is)
(NP (NP (DT a) (JJ linear) (NN operator)) (, ,)
(SBAR (S (NP (PRP it))
(VP (MD can)
(VP (AUX be)
(VP (VBN described)
(PP (IN by)
(NP (NNP matrix) (NN multiplication))))
(. .)))
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.
(Parse)
(S1 (S (PP (IN By)
(NP (NP (NN analogy))
(PP (IN with)
(NP (NP (DT the) (NN term) ('' '') (NN eigenvector)) ('' '')
(PP (IN for)
(NP (NP (DT a) (NN vector))
(SBAR (WHNP (WDT which)) (, ,)
(S (SBAR (WHADVP (WRB when))
(S (VP (VBN operated) (PRT (RP on))
(PP (IN by)
(NP (DT a) (JJ linear) (NN operator)))
(VP (AUX is)
(NP (NP (DT another) (NN vector))
(SBAR (WHNP (WDT which))
(S (VP (AUX is)
(NP (NP (DT a) (JJ scalar) (NN multiple))
(PP (IN of)
(NP (PRP itself)))))
(, ,)
(NP (DT the) (NN term) ('' '') (NN eigenplane) ('' ''))
(VP (MD can)
(VP (AUX be)
(VP (VBN used)
(S (VP (TO to)
(VP (VB describe)
(S (NP (NP (DT a) (CD two)) (: -)
(SBAR (S (NP (JJ dimensional) (NN plane) (PRN (-LRB- -LRB-)
(NP (DT a) ('' '') (CD 2)) (: -)
(NP (NN plane)) ('' '') (-RRB- -RRB-) (, ,)
(PP (JJ such) (IN that)
(NP (NP (DT the) (NN operation))
(PP (IN of)
(NP (NP (DT a) (JJ linear) (NN operator))
(PP (IN on)
(NP (NP (DT a) (NN vector))
(PP (IN in)
(NP (DT the) (CD 2)))
(: -)) (NN plane)) (ADVP (RB always))
(VP (VBZ yields)
(NP (NP (DT another) (NN vector))
(PP (IN in)
(NP (DT the) (JJ same) (CD 2))))
(: -))
(NP (NN plane)))
(. .)))
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.
(Parse)
(S1 (S (PP (IN In)
(NP (NN fact)))
(, ,)
(SBAR (IN if)
(S (NP (DT a) (JJ linear) (NN operator))
(VP (AUX has)
(NP (DT the) (NN orthogonal)))))
(NP (NP (NNP matrix)) (CC and)
(NP (DT this) (NNP matrix)))
(VP (AUX is)
(ADJP (JJ symmetric)
(SBAR (S (NP (DT the) ('' '')
(NP (JJ Symmetric) (NN operator) (NN theorem)) ('' '')
(PP (JJ valid) (IN in)
(NP (NP (JJ -LSB-) (NN Formula))
(NP (CD 40) (NN -RSB-))))
(PRN (-LRB- -LRB-)
(NP (DT any) (NN dimension)) (-RRB- -RRB-)))
(VP (VBZ applies)
(S (VP (VBG saying)
(SBAR (IN that)
(S (NP (PRP it))
(VP (AUX has)
(VP ('' '')
(NP (NNP n) ('' '') (JJ orthogonal) (NNS eigenvectors))))
(. .)))
In Wikipedia:
Let T be a linear operator on a vector space V, and let [Formula 146] be distinct eigenvalues of T.
(Parse)
(S1 (S (S (VP (VB Let)
(S (NP (NNS T))
(VP (AUX be)
(NP (NP (DT a) (JJ linear) (NN operator))
(PP (IN on)
(NP (DT a) (NN vector) (NN space) (NN V)))))
(, ,) (CC and)
(S (VP (VB let)
(S (NP (NP (JJ -LSB-) (NN Formula))
(NP (CD 146) (NNS -RSB-)))
(VP (AUX be)
(NP (NP (JJ distinct) (NNS eigenvalues))
(PP (IN of)
(NP (NNP T)))))
(. .)))
In Wikipedia:
Let [Formula 40] be a second degree linear operator.
(Parse)
(S1 (S (VP (VB Let)
(S (NP (NP (JJ -LSB-) (NN Formula))
(NP (CD 40) (NN -RSB-)))
(VP (AUX be)
(NP (NP (DT a) (JJ second) (NN degree))
(NP (JJ linear) (NN operator)))
(. .)))
In Wikipedia:
In linear algebra, an involution is a linear operator “T” such that [Formula 2].
(Parse)
(S1 (S (PP (IN In)
(NP (JJ linear) (NN algebra)))
(, ,)
(NP (DT an) (NN involution))
(VP (AUX is)
(NP (NP (DT a) (JJ linear) (NN operator)) ('' '')
(VP (VBG T) ('' '')
(NP (NP (PDT such) (DT that) (JJ -LSB-) (NN Formula))
(ADJP (CD 2) (JJ -RSB-)))
(. .)))
In Wikipedia:
Recall that an eigenvector of a linear operator “A” is a (non – zero) vector “x” such that “Ax” = “x” for some scalar.
(Parse)
(S1 (S (VP (VB Recall)
(SBAR (IN that)
(S (NP (NP (DT an) (NN eigenvector))
(PP (IN of)
(NP (DT a) (JJ linear) (NN operator)))
('' '')) (DT A) ('' '')
(VP (AUX is)
(NP (NP (DT a) (PRN (-LRB- -LRB-)
(NP (NP (NN non)) (: -)
(NP (CD zero)))
(-RRB- -RRB-)) (NN vector) ('' ''))
(VP (SYM x) ('' '')
(ADJP (JJ such)
(SBAR (IN that)
(S ('' '')
(NP (NN Ax)) ('' '')
(VP (SYM =)
(NP (NNP ) (NNP ) ('' '') (NNP x)) ('' '')
(PP (IN for)
(NP (DT some) (JJ scalar) (CD ) (NNS ))))
(. .)))
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.
(Parse)
(S1 (S (SBAR (RB Just) (IN as)
(S (NP (PRP you))
(VP (MD would)
(VP (VB expect)
(PP (IN from)
(NP (NP (DT the) (NN theory))
(PP (IN of)
(NP (NP (DT the) (NNS eigenvalues))
(PP (IN of)
(NP (DT an) (JJ ordinary) (JJ linear) (NN operator))))
(PP (IN on)
(NP (DT a) (CD four) (JJ dimensional) (NN vector) (NN space))))
(, ,)
(NP (NP (DT the) (JJ different) (NNS types))
(PP (IN of)
(NP (NNP Weyl) (NN tensor)))
(PRN (-LRB- -LRB-)
(PP (IN at)
(NP (DT a) (VBN given) (NN event)))
(-RRB- -RRB-)))
(VP (MD can)
(VP (AUX be)
(VP (VBN determined)
(PP (IN by)
(S (VP (VBG solving)
(NP (DT a) (JJ certain) (JJ quartic) (NN polynomial)))))
(. .)))
In Wikipedia:
The operator [Formula 6] is linear.
(Parse)
(S1 (S (NP (NP (DT The) (NN operator) (JJ -LSB-) (NN Formula))
(ADJP (CD 6) (JJ -RSB-)))
(VP (AUX is)
(ADJP (JJ linear)))
(. .)))
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”.
(Parse)
(S1 (S (NP (NP (DT The) (NN sum))
(PP (IN of)
(NP (DT the) ('' '')
(NP (NP (NNP k) ('' '') (JJS largest) (JJ singular) (NNS values))
(PP (IN of) (PRN ('' '')
(S (NP (NNP M)) ('' '')
(VP (AUX is)
(NP (DT a) (NNP matrix) (NN norm))))
(, ,))
(NP (DT the) (NNP Ky) (NN Fan))))
('' '')
(SBAR (S (NP (NNP k) ('' '') (PRN (: -)
(NP (NP (NN norm))
(PP (IN of) ('' '')
(NP (NNP M) ('' '')
(NP (NP (DT The) (JJ first))
(PP (IN of)
(NP (DT the) (NNP Ky) (NN Fan) (NNS norms))))
(, ,)
(NP (DT the) (NNP Ky) (NN Fan) (CD 1)))))
(: -)) (NN norm))
(VP (AUX is)
(NP (NP (DT the) (JJ same))
(PP (IN as)
(NP (NP (DT the) (NN operator) (NN norm))
(PP (IN of) ('' '')
(NP (NNP M)))
('' '')
(PP (IN as)
(NP (NP (DT a) (JJ linear) (NN operator))
(PP (IN with)
(NP (NP (NN respect))
(PP (TO to)
(NP (DT the) (NNP Euclidean) (NNS norms)))))
(PP (IN of) ('' '')
(NP (NN K)))))
(VP (AUX 'm) (UCP ('' '') (CC and) ('' '')
(NP (NNP K)) (CC 'n) ('' '')))
(. .)))
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.
(Parse)
((S (SBAR (SBAR (IN If)
(S (NP (NN T))
(VP (VBZ is)
(NP (DT a) (JJ linear) (NN operator)
(PP (IN on)
(NP (DT an) (JJ arbitrary) (NN vector) (NN space))))
(CC and)
(SBAR (IN if)
(S (NP (EX there))
(VP (VBZ is)
(NP (DT a) (JJ monic) (NN polynomial) (NN P)) (X (JJ such) (IN that)
(NP (SYM A[fj]))))
(, ,) (ADVP (RB then))
(S (NP (NNS parts) (-LRB- -LRB-) (NN A) (-RRB- -RRB-) (CC and) (-LRB- -LRB-) (NN B) (-RRB- -RRB-)
(PP (IN of)
(NP (NN Theorem) (CD 12))))
(VP (VBP are)
(ADJP (JJ valid)
(PP (IN for)
(NP (NN T)))))
(PP (IN with)
(NP (NP (DT the) (NN proof))
(SBAR (NP (WDT which))
(S (NP (PRP we))
(VP (VBD gave)
(NP (-NONE- T))))))
(. .))
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).
(Parse)
(S1 (S (S (S (VP (TO To)
(VP (VB formalize)
(NP (PRP it)))))
(NP (PRP we))
(VP (AUX need)
(S (VP (TO to)
(VP (VB say)))
(: :)
(S (NP (VB let) ('' '') (NN T) ('' ''))
(VP (AUX be)
(NP (NP (DT a) (JJ linear) (NN operator))
(VP (VBN defined)
(PP (IN on)
(NP (NP (DT a) (NN family) ('' '') (NN F)) ('' '')
(PP (IN of)
(NP (NP (NNS functions))
(SBAR (WHNP (WDT which))
(S (VP (AUX is)
(ADJP (JJ dense)
(PP (IN in)
(NP (NP (NP (DT both) (JJ -LSB-) (NN Formula))
(NP (CD 1) (NN -RSB-)))
(CC and)
(NP (NP (NP (JJ -LSB-) (NN Formula))
(NP (CD 2) (NN -RSB-)))
(PRN (-LRB- -LRB-)
(PP (IN for)
(NP (NN example)))
(, ,)
(NP (NP (DT the) (NN family))
(PP (IN of)
(NP (DT all) (JJ simple) (NNS functions))))
(-RRB- -RRB-)))))
(. .)))
In Wikipedia:
Kernel (linear operator).
(Parse)
(S1 (NP (NP (NNP Kernel)) (PRN (-LRB- -LRB-)
(NP (JJ linear) (NN operator)) (-RRB- -RRB-)) (. .)))
And 259 more sentences.