Learned by Reading
In Wikipedia:
Every element “x” of “g” defines the adjoint endomorphism “ad” (“x”) (also written as “ad ’x”) of “g” with the help of the Lie bracket, as Now, supposing “g” is of finite dimension, the trace of the composition of two such endomorphisms defines a symmetric bilinear form with values in “K”, the Killing form’ “on” g”.
(Parse)
(S1 (S (NP (DT Every) (NN element)) ('' '')
(NP (SYM x) ('' '')
(PP (IN of) ('' '')
(NP (NNP g))))
('' '')
(VP (VBZ defines)
(NP (NP (DT the) (JJ adjoint) (NN endomorphism) ('' '') (NN ad) ('' '') (PRN (-LRB- -LRB-) (FRAG ('' '')
(NP (SYM x)) ('' '')) (-RRB- -RRB-) (-LRB- -LRB-)
(VP (ADVP (RB also)) (VBN written)
(PP (IN as) ('' '')
(NP (FW ad) (FW 'x) ('' ''))))
(-RRB- -RRB-)))
(PP (IN of) ('' '')
(NP (NNP g)) ('' '')))
(PP (IN with)
(NP (NP (DT the) (NN help))
(PP (IN of)
(NP (DT the) (NN Lie) (NN bracket)))))
(, ,) (ADVP (RB as) (RB Now)) (, ,)
(S (VP (VBG supposing)
(S ('' '')
(S (NP (NNP g)) ('' '')
(VP (AUX is)
(PP (IN of)
(NP (JJ finite) (NN dimension)))))
(, ,)
(NP (DT the))
(VP (VB trace)
(PP (IN of)
(NP (NP (DT the) (NN composition))
(PP (IN of)
(NP (NP (CD two) (JJ such) (NNS endomorphisms))
(SBAR (S (VP (VBZ defines)
(NP (NP (DT a) (JJ symmetric) (NN bilinear) (NN form))
(PP (IN with)
(NP (NNS values))))
(PP (IN in) ('' '')
(NP (NNP K) ('' '') (, ,)
(NP (DT the) (VBG Killing) (NN form) (POS '))))
('' '')
(PP (IN on) ('' '')
(NP (NNP g)) ('' '')))))
(. .)))
In Wikipedia:
We say that two symmetric bilinear forms are equivalent’ “if one can be obtained from the other by adding zero or more copies of a hyperbolic plane (the non – degenerate two – dimensional symmetric bilinear form with a norm 0 vector).
(Parse)
(S1 (S (NP (PRP We))
(VP (VBP say)
(SBAR (IN that)
(S (NP (CD two) (JJ symmetric) (JJ bilinear) (NNS forms))
(VP (AUX are)
(NP (NN equivalent) ('' ')) ('' '')
(SBAR (IN if)
(S (NP (PRP one))
(VP (MD can)
(VP (AUX be)
(VP (VBN obtained)
(PP (IN from)
(NP (DT the) (JJ other)))
(PP (IN by)
(S (VP (VBG adding)
(NP (NP (QP (CD zero) (CC or) (JJR more)) (NNS copies))
(PP (IN of)
(NP (NP (DT a) (JJ hyperbolic) (NN plane)) (PRN (-LRB- -LRB-)
(NP (NP (DT the) (NN non)) (: -)
(NP (VB degenerate) (CD two)) (: -)
(NP (NP (JJ dimensional) (JJ symmetric) (NN bilinear) (NN form))
(PP (IN with)
(NP (DT a) (NN norm)))
(NP (CD 0) (NN vector))))
(-RRB- -RRB-)))))
(. .)))
In Wikipedia:
Thus “B” is a positive definite symmetric bilinear form, in other words, an inner product on “V”.
(Parse)
(S1 (S (ADVP (RB Thus)) ('' '')
(NP (SYM B)) ('' '')
(VP (AUX is)
(NP (NP (DT a) (JJ positive) (JJ definite) (JJ symmetric) (NN bilinear) (NN form)) (, ,)
(PP (IN in)
(NP (JJ other) (NNS words)))
(, ,)
(NP (NP (DT an) (JJ inner) (NN product))
(PP (IN on) ('' '')
(NP (NNP V) ('' '')))
(. .)))
In Wikipedia:
A map [Formula 1] is a symmetric bilinear form on the space if: The last two axioms only imply linearity in the first argument, but the first immediately implies linearity in the second argument then too.
(Parse)
(S1 (S (S (NP (NP (DT A) (NN map) (NN -LSB-) (NN Formula))
(NP (CD 1) (NN -RSB-)))
(VP (AUX is)
(NP (NP (DT a) (JJ symmetric) (NN bilinear) (NN form))
(PP (IN on)
(NP (NP (DT the) (NN space)) (ADVP (IN if))))
(: :)
(S (S (NP (DT The) (JJ last) (CD two) (NNS axioms)) (ADVP (RB only))
(VP (VBP imply)
(NP (NN linearity))
(PP (IN in)
(NP (DT the) (JJ first) (NN argument)))))
(, ,) (CC but)
(S (NP (DT the) (JJ first)) (ADVP (RB immediately))
(VP (VBZ implies)
(NP (NP (NN linearity))
(PP (IN in)
(NP (NP (DT the) (JJ second) (NN argument)) (ADVP (RB then)))))
(ADVP (RB too)))))
(. .)))
In Wikipedia:
Here ““is a nondegenerate skew symmetric bilinear form where F ’” is the field of scalars.
(Parse)
(S1 (S (ADVP (RB Here)) ('' '')
(NP (NN )) ('' '')
(VP (AUX is)
(NP (NP (DT a) (JJ nondegenerate) (JJ skew) (JJ symmetric) (NN bilinear) (NN form))
(SBAR (WHADVP (WRB where))
(S (NP (SYM F) ('' ') ('' ''))
(VP (AUX is)
(NP (NP (DT the) (NN field))
(PP (IN of)
(NP (NNS scalars)))
(. .)))
In Wikipedia:
The associated symmetric bilinear form on “C” (“V”, “Q”) is given by One can check that this reduces to the original bilinear form when restricted to “V”.
(Parse)
(S1 (S (NP (NP (DT The) (VBN associated) (JJ symmetric) (NN bilinear) (NN form))
(PP (IN on) ('' '')
(NP (NP (NNP C) ('' '') (NNS )) (-LRB- -LRB-) ('' '')
(NP (CD V)) ('' '') (PRN (, ,) ('' '') (NNP Q) ('' '') (-RRB- -RRB-)))))
(VP (AUX is)
(VP (VBN given)
(PP (IN by)
(NP (NP (NN One))
(SBAR (S (VP (MD can)
(VP (VB check)
(SBAR (IN that)
(S (NP (DT this))
(VP (VBZ reduces)
(PP (TO to)
(NP (DT the) (JJ original) (JJ bilinear) (NN form)))
(SBAR (WHADVP (WRB when))
(S (VP (VBN restricted)
(PP (TO to) ('' '')
(NP (CD V)) ('' ''))))
(. .)))
In Wikipedia:
A metric tensor is defined to be a nondegenerate symmetric bilinear form on each tangent space, which varies smoothly from point to point.
(Parse)
(S1 (S (NP (DT A) (JJ metric) (NN tensor))
(VP (AUX is)
(VP (VBN defined)
(S (VP (TO to)
(VP (AUX be)
(NP (NP (DT a) (JJ nondegenerate) (JJ symmetric) (NN bilinear) (NN form))
(PP (IN on)
(NP (DT each) (JJ tangent) (NN space)))
(, ,)
(SBAR (WHNP (WDT which))
(S (VP (VBZ varies) (ADVP (RB smoothly))
(PP (PP (IN from)
(NP (NN point)))
(PP (TO to)
(NP (NN point))))
(. .)))
In Wikipedia:
So, an inner product on a real vector space is a “positive – definite symmetric bilinear form”.
(Parse)
(S1 (S (ADVP (RB So)) (, ,)
(NP (NP (DT an) (JJ inner) (NN product))
(PP (IN on)
(NP (DT a) (JJ real) (NN vector) (NN space))))
(VP (AUX is)
(NP (NP (DT a) ('' '') (NN positive)) (: -)
(NP (JJ definite) (JJ symmetric) (NN bilinear) (NN form) ('' ''))))
(. .)))
In Wikipedia:
In mathematics, a quasi – Frobenius Lie algebra’ “over a field [Formula 2] is a Lie algebra equipped with a nondegenerate skew – symmetric bilinear form for all [Formula 8], [Formula 9], [Formula 10] in [Formula 5].
(Parse)
(S1 (S (PP (IN In)
(NP (NNS mathematics)))
(, ,)
(NP (NP (DT a) (NNS quasi)) (: -)
(NP (NNP Frobenius)))
(VP (VBP Lie)
(NP (NP (NP (NP (NNS algebra) (POS ')) ('' '')
(PP (IN over)
(NP (DT a) (NN field))))
(JJ -LSB-) (NN Formula))
(SBAR (S (NP (CD 2) (NN -RSB-))
(VP (AUX is)
(NP (NP (DT a) (NN Lie) (NN algebra))
(VP (VBN equipped)
(PP (IN with)
(NP (DT a) (JJ nondegenerate) (NN skew)))))
(: -)
(NP (NP (NP (NP (JJ symmetric) (NN bilinear) (NN form))
(PP (IN for)
(NP (DT all) (JJ -LSB-) (NN Formula)))
(NP (CD 8) (NN -RSB-)))
(, ,)
(NP (NP (JJ -LSB-) (NN Formula))
(NP (CD 9) (NN -RSB-)))
(, ,)
(NP (JJ -LSB-) (NN Formula)))
(NP (CD 10) (NNS -RSB-))))
(PP (IN in)
(NP (NP (JJ -LSB-) (NN Formula))
(NP (CD 5) (NN -RSB-)))))
(. .)))
In Wikipedia:
A GUT model basically consists of a gauge group which is a compact Lie group, a connection form for that Lie group, a Yang – Mills action for that connection given by an invariant symmetric bilinear form over its Lie algebra (which is specified by a coupling constant for each factor), a Higgs sector consisting of a number of scalar fields taking on values within real / complex representations of the Lie group and chiral Weyl fermions taking on values within a complex rep of the Lie group.
(Parse)
(S1 (S (S (NP (DT A) (NN GUT) (NN model)) (ADVP (RB basically))
(VP (VBZ consists)
(PP (IN of)
(NP (NP (DT a) (NN gauge) (NN group))
(SBAR (WHNP (WDT which))
(S (VP (AUX is)
(NP (NP (DT a) (JJ compact) (NN Lie) (NN group)) (, ,)
(NP (NP (DT a) (NN connection) (NN form))
(PP (IN for)
(NP (NP (DT that) (NN Lie) (NN group)) (, ,)
(NP (DT a) (NNP Yang))))
(: -)
(S (NP (NP (NNP Mills) (NN action))
(PP (IN for)
(NP (DT that) (NN connection))))
(VP (VBN given)
(PP (IN by)
(NP (NP (DT an) (JJ invariant) (JJ symmetric) (NN bilinear) (NN form))
(PP (IN over)
(NP (NP (PRP$ its) (NN Lie) (NN algebra)) (PRN (-LRB- -LRB-)
(SBAR (WHNP (WDT which))
(S (VP (AUX is)
(VP (VBN specified)
(PP (IN by)
(NP (NP (DT a) (NN coupling) (JJ constant))
(PP (IN for)
(NP (DT each) (NN factor)))
(-RRB- -RRB-)) (, ,)
(NP (NP (DT a) (NNP Higgs) (NN sector))
(VP (VBG consisting)
(PP (IN of)
(S (NP (NP (DT a) (NN number))
(PP (IN of)
(NP (JJ scalar) (NNS fields))))
(VP (VBG taking) (PRT (RP on))
(NP (NNS values))
(PP (IN within)
(NP (NP (JJ real) (JJ /) (JJ complex) (NNS representations))
(PP (IN of)
(NP (NP (DT the) (NN Lie) (NN group)) (CC and)
(NP (NP (JJ chiral) (NNP Weyl) (NNS fermions))
(VP (VBG taking)
(PP (IN on)
(NP (NP (NNS values))
(PP (IN within)
(NP (NP (DT a) (JJ complex) (NN rep))
(PP (IN of)
(NP (DT the) (NN Lie) (NN group)))
(. .)))
In Wikipedia:
Taking the standard symmetric bilinear form with orthonormal basis “v ’i”, the map sending a lattice to its dual lattice gives an automorphism with square the identity, giving the permutation that sends each label to its negative modulo “n”.
(Parse)
(S1 (S (S (VP (VBG Taking)
(NP (NP (DT the) (JJ standard) (JJ symmetric) (NN bilinear) (NN form))
(PP (IN with)
(NP (JJ orthonormal) (NN basis) ('' '') (JJ v) (NNS 'i)))
('' ''))))
(, ,)
(NP (NP (DT the) (NN map))
(VP (VBG sending)
(NP (DT a) (NN lattice))
(PP (TO to)
(NP (PRP$ its) (JJ dual) (NN lattice)))))
(VP (VBZ gives)
(NP (NP (DT an) (NN automorphism))
(PP (IN with)
(NP (NN square))))
(NP (DT the) (NN identity)) (, ,)
(S (VP (VBG giving)
(NP (NP (DT the) (NN permutation) (NN ))
(SBAR (WHNP (WDT that))
(S (VP (VBZ sends)
(NP (DT each) (NN label))
(PP (TO to)
(NP (PRP$ its) (JJ negative) (NN modulo) ('' '') (NNP n) ('' ''))))
(. .)))
In Wikipedia:
For example, let (“V”,) be a finite dimensional real symplectic vector space (so is a nondegenerate skew symmetric bilinear form on “V”).
(Parse)
(S1 (S (PP (IN For)
(NP (NN example)))
(, ,)
(VP (VB let)
(S (VP (PRN (-LRB- -LRB-) (FRAG (FRAG ('' '')
(NP (CD V)) ('' '')) (, ,) (INTJ (UH )))
(-RRB- -RRB-)) (AUX be)
(NP (NP (DT a) (JJ finite) (JJ dimensional) (JJ real) (JJ symplectic) (NN vector) (NN space)) (PRN (-LRB- -LRB-)
(S (NP (RB so) (JJ ))
(VP (AUX is)
(NP (NP (DT a) (JJ nondegenerate) (JJ skew) (JJ symmetric) (NN bilinear) (NN form))
(PP (IN on) ('' '')
(NP (NNP V)))
('' '') (-RRB- -RRB-)))
(. .)))
In Wikipedia:
An orthogonal basis is one such that where <, > is the symmetric bilinear form associated to “Q”.
(Parse)
(S1 (S (NP (DT An) (JJ orthogonal) (NN basis))
(VP (AUX is)
(NP (NP (CD one))
(ADJP (ADJP (JJ such))
(SBAR (IN that)
(S (SBAR (WHADVP (WRB where))
(S (NP (NNP <))
(VP (VBZ ))))
(, ,)
(NP (JJ ) (NN >))
(VP (AUX is)
(NP (NP (DT the) (JJ symmetric) (NN bilinear) (NN form))
(VP (VBN associated)
(S (VP (TO to)
(VP ('' '')
(NP (NNP Q))))
('' ''))))
(. .)))
In Wikipedia:
Explicitly, the metric is a symmetric bilinear form on each tangent space of “M” which varies in a smooth (or differentiable) manner from point to point.
(Parse)
(S1 (S (ADVP (RB Explicitly)) (, ,)
(NP (DT the) (NN metric))
(VP (AUX is)
(NP (NP (DT a) (JJ symmetric) (NN bilinear) (NN form))
(PP (IN on)
(NP (DT each) (JJ tangent) (NN space)))
(PP (IN of) ('' '')
(NP (NNP M) ('' '')
(SBAR (WHNP (WDT which))
(S (VP (VBZ varies)
(PP (IN in)
(NP (DT a) (JJ smooth) (PRN (-LRB- -LRB-) (CC or) (JJ differentiable) (-RRB- -RRB-)) (NN manner)))
(PP (PP (IN from)
(NP (NN point)))
(PP (TO to)
(NP (NN point)))))
(. .)))
In Wikipedia:
The signature’ “of a metric tensor (or more generally a nondegenerate symmetric bilinear form, thought of as quadratic form) is the number of positive and negative eigenvalues of the metric.
(Parse)
(S1 (S (NP (NP (DT The) (NN signature) (POS ')) ('' '')
(PP (IN of)
(NP (DT a) (JJ metric) (NN tensor)))
(PRN (-LRB- -LRB-) (CC or) (ADVP (RBR more) (RB generally))
(NP (NP (DT a) (JJ nondegenerate) (JJ symmetric) (NN bilinear) (NN form)) (, ,)
(VP (VBN thought)
(PP (IN of))
(PP (IN as)
(NP (JJ quadratic) (NN form)))))
(-RRB- -RRB-)))
(VP (AUX is)
(NP (NP (DT the) (NN number))
(PP (IN of)
(NP (NP (ADJP (JJ positive) (CC and) (JJ negative)) (NNS eigenvalues))
(PP (IN of)
(NP (DT the) (JJ metric))))
(. .)))
In Wikipedia:
If “B” is any symmetric bilinear form on a vector space, and “Q” is the quadratic form defined by These formulas even apply in the case where the field of scalars has characteristic two, though the left – hand sides are all zero in this case.
(Parse)
(S1 (S (SBAR (IN If)
(S (S ('' '')
(NP (SYM B)) ('' '')
(VP (AUX is)
(NP (NP (DT any) (JJ symmetric) (NN bilinear) (NN form))
(PP (IN on)
(NP (DT a) (NN vector) (NN space)))
(, ,) (CC and)
(S ('' '')
(NP (NNP Q)) ('' '')
(VP (AUX is)
(NP (NP (DT the) (JJ quadratic) (NN form))
(VP (VBN defined)
(PP (IN by)
(NP (DT These) (NNS formulas) (RB even)))
(S (VP (VB apply)
(PP (IN in)
(NP (NP (DT the) (NN case))
(SBAR (WHADVP (WRB where))
(S (NP (NP (DT the) (NN field))
(PP (IN of)
(NP (NNS scalars))))
(VP (AUX has)
(ADJP (JJ characteristic)
(NP (CD two)) (, ,)
(PP (IN though)
(NP (DT the) (NN left)))))
(: -)
(NP (NN hand) (NNS sides))
(VP (AUX are)
(NP (NP (DT all) (CD zero))
(PP (IN in)
(NP (DT this) (NN case)))))
(. .)))
In Wikipedia:
The quadratic form defining the Plcker relation comes from a symmetric bilinear form of signature (3,3).
(Parse)
(S1 (S (NP (NP (DT The) (JJ quadratic) (NN form))
(PP (VBG defining)
(NP (DT the) (NNP Plcker) (NN relation))))
(VP (VBZ comes)
(PP (IN from)
(NP (NP (DT a) (JJ symmetric) (NN bilinear) (NN form))
(PP (IN of)
(NP (NP (NN signature)) (PRN (-LRB- -LRB-)
(NP (CD 3,3)) (-RRB- -RRB-))))
(. .)))
In Wikipedia:
The metric “g” is defined to be the real part of “h”: The form “g” is a symmetric bilinear form on “TM” C, the complexified tangent bundle.
(Parse)
(S1 (S (S (NP (DT The) (JJ metric) ('' '') (NNP g) ('' ''))
(VP (AUX is)
(VP (VBN defined)
(S (VP (TO to)
(VP (AUX be)
(NP (NP (DT the) (JJ real) (NN part))
(PP (IN of) (FRAG ('' '')
(NP (NNP h)) ('' ''))))
(: :)
(S (NP (DT The) (NN form) ('' '') (NNP g) ('' ''))
(VP (AUX is)
(NP (NP (DT a) (JJ symmetric) (NN bilinear) (NN form))
(PP (IN on) ('' '')
(NP (NP (NNP TM) ('' '') (NNP C)) (, ,)
(NP (DT the) (JJ complexified) (NN tangent) (NN bundle))))
(. .)))
In Wikipedia:
Given any unimodular symmetric bilinear form over the integers, “Q”, there is a simply – connected closed 4 – manifold “M” with intersection form “Q”.
(Parse)
(S1 (S (PP (VBN Given)
(NP (NP (DT any) (JJ unimodular) (JJ symmetric) (NN bilinear) (NN form))
(PP (IN over)
(NP (DT the) (NNS integers)))
(, ,) ('' '')
(NP (NNP Q)) ('' '')))
(, ,)
(NP (EX there))
(VP (AUX is)
(VP (X (DT a) (RB simply)) (: -)
(VP (VBN connected)
(ADJP (JJ closed)
(NP (CD 4))))
(: -)
(NP (NP (JJ manifold) ('' '') (NNP M)) ('' '')
(PP (IN with)
(NP (NN intersection) (NN form) ('' '') (NNP Q) ('' '')))
(. .)))
And 17 more sentences.