Then the cotangent space at x is defined as the dual space of T < sub > x </ sub > M:

Then I and I < sup > 2 </ sup > are real vector spaces, and T < sub > x </ sub > M may be defined as the dual space of the quotient space I / I < sup > 2 </ sup >.

Let X be a normed topological vector space over F, compatible with the absolute value in F. Then in X *, the topological dual space X of continuous F-valued linear functionals on X, all norm-closed balls are compact in the weak -* topology.

Then the continuous dual of X is the set and bounded subsets of X are norm-bounded, hence the Banach space is the strong dual of X.

Then ba ( Σ ) = B ( Σ )* is the continuous dual space of B ( Σ ).

Then with the 1971 Oregon v. Mitchell decision, the Supreme Court ruled that state and local elections did not have to abide by the lowered voting age, though there would have to be dual elections in the 47 states where the lower federal voting age was not valid.

Then ρ induces a representation ρ < sup >*</ sup > on the dual of the vector space Hom ( C < sup > n </ sup >, C ).

Then the transpose ( or dual ) of T is an operator satisfying:

Then Splendid erupts to about 250 feet, The dual eruptions last from 30 to 120 minutes.

For f ∈ R, define D < sub > f </ sub > to be the set of ideals of R not containing f. Then each D < sub > f </ sub > is an open subset of Spec ( R ), and is a basis for the Zariski topology.

Then in 1991, President George H. W. Bush made an attempt to abolish affirmative action altogether claiming that “ any regulation, rule, enforcement practice or other aspect of these programs that mandates, encourages, or otherwise involves the use of quotas, preferences, set-asides or other devices on the basis of race, sex, religion or national origin are to be terminated as soon as is legally feasible.

Then the columns of V such that the corresponding form an orthonormal basis of the nullspace of A.

Then, in 1920, his father hired tutors to teach him the four books, considered the basis of all Chinese culture.

Then in 1913, using this formula as a basis, Albert Einstein and Otto Stern published a paper of great significance in which they suggested for the first time the existence of a residual energy that all oscillators have at absolute zero.

Then, in 1980, Digital, Intel and Xerox ( the DIX consortium ) published an open standard for Ethernet that was soon adopted as the basis of standardization by the IEEE and the ISO.

Then he would break from the Democrats and form a third party using the Share Our Wealth plan as its basis.

Then, the B < sub > m </ sub > make up a neighbourhood basis of 0 in K.

Then, in 1870, Congress passed a law that prohibited the use of a Presidential pardon as the basis for claiming sale proceeds, and further said that acceptance of such a pardon was evidence that the person pardoned did provide support to the South and was ineligible to recover sale proceeds.

Then, in 1863, the college instituted the policy that no applicant was to be rejected on the basis of race.

In addition to the two songs released, two other Lennon demos ( which have been bootlegged ) became the basis for additional work by the other three Beatles: " Now and Then " and " Grow Old With Me ".

Then the contravariant coordinates of any vector v can be obtained by the dot product of v with the contravariant basis vectors:

Then there is a cyclic cubic field inside the cyclotomic field of pth roots of unity, and a normal integral basis of periods for the integers of this field ( an instance of the Hilbert – Speiser theorem ).

Then, prior to their season opening game of the 1963 season, Saskatchewan acquired Ottawa Rough Riders quarterback Ron Lancaster on July 30 on a straight cash basis following three years with the Eastern Riders.

Then there is an orthonormal basis

Then applying the Gram – Schmidt process to the three vectors ( A < sub > 2 </ sub >− A < sub > 1 </ sub >, A < sub > 3 </ sub >− A < sub > 1 </ sub >, V ) produces an orthonormal basis of space, the third vector of which will be normal to plane A.

Then the judge can sustain ( rule in favor of ) a demurrer on the basis that the complaint's date-related allegations indicate it was filed too late (" the statute of limitations has run "), unless the plaintiff can show a typographical error ( a so-called " scrivener's error ") occurred in the drafting of the complaint.

Then labeling the basis elements of V by 0, 1, 2, ..., 22, ∞, W can be defined as the span of the words S < sub > t </ sub > together with the word consisting of all basis vectors.

Then a new pair of basis points is selected, and the process is repeated.

Then from the known formulae you can calculate the basis vectors of the reciprocal lattice.

Then h is injective on the set of canonical monomials and its range is a basis of the K-vector space U ( L ).

Then for every there is a unique linear combination of the basis vectors that equals v:

Then the bit flip code from above can recover by transforming into the Hadamard basis before and after transmission through.

Then the statistician must analyze the properties of and, which are viewed as random vectors since a randomly different selection of n cases to observe would have resulted in different values for them.

Let vectors and let denote the matrix with elements of a and b. Then the area of the parallelogram generated by a and b is equal to.

Let vectors and let Then the area of the parallelogram generated by a and b is equal to.

Then express the seed ( the initial conditions of the LRS ) as a linear combination of the eigenbasis vectors:

Then can be broken down to its column vectors as follows:

Then one seeks vectors maximizing the same correlation subject to the constraint that they are to be uncorrelated with the first pair of canonical variables ; this gives the second pair of canonical variables.

Then this inner product between f and g can be roughly understood as the dot product between approximating vectors and, in the limit as the number of sampling points goes to infinity.

Then the pair of vectors W =( F, R × F ) is called a wrench.

Then the inner product of two tangent vectors is

Then the locus of points at the end of such vectors is called the radial of the curve.

: Then the kernel of s is the one-dimensional subspace consisting of all vectors ( x < sub > 1 </ sub >, 0, 0, ...).

Then their respective planes are perpendicular to vectors a and b, and the direction of L must be perpendicular to both.

Then the general expression of a structural form is, where f is a function, possibly from vectors to vectors in the case of a multiple-equation model.

# A " Min-max " Collatz – Wielandt formula takes a form similar to the one above: for all strictly positive vectors x, let g ( x ) be the maximum value of < sub > i </ sub > / x < sub > i </ sub > taken over i. Then g is a real valued function whose minimum is the Perron – Frobenius eigenvalue.

Fix a point p ∈ M, and an orthonormal basis X < sub > 1 </ sub >, X < sub > 2 </ sub > of tangent vectors at p. Then the principal curvatures are the eigenvalues of the symmetric matrix

) Then a linear transformation for such vectors can be defined by a square matrix with one row and one column for each element of the vector.

Then, any two-form can be expressed as a linear combination of tensor products of pairs of vectors, especially a linear combination of tensor products of pairs of basis vectors.