linear

`` When working from one of my sketches I square it up and project its linear form freehand to the watercolor sheet with charcoal.

When this linear draft is completed, I dust it down to a faint image.

The theory predicts a linear dependence of Af on Af, where J is the experimentally determined Curie-Weiss constant.

Theoretical studies of the hydrogen bond generally agree that the Af bond will be linear in the absence of peculiarities of packing in the solid.

If the Af bond is linear then there are three reasonable positions for the hydrogen atoms: ( 1 ) The hydrogen atoms are centered and hence all lie on a sheet midway between the oxygen sheets ; ;

However, one must expect that vaporization and ejection of material by hypervelocity impacts would cause a deviation from a linear relationship.

We are trying to study a linear operator T on the finite-dimensional space V, by decomposing T into a direct sum of operators which are in some sense elementary.

Let T be a linear operator on the finite-dimensional vector space V over the field F.

If Af are the projections associated with the primary decomposition of T, then each Af is a polynomial in T, and accordingly if a linear operator U commutes with T then U commutes with each of the Af, i.e., each subspace Af is invariant under U.

Let N be a linear operator on the vector space V.

Let T be a linear operator on the finite-dimensional vector space V over the field F.

Suppose that the minimal polynomial for T decomposes over F into a product of linear polynomials.

Then every linear operator T in V can be written as the sum of a diagonalizable operator D and a nilpotent operator N which commute.

From these results, one sees that the study of linear operators on vector spaces over an algebraically closed field is essentially reduced to the study of nilpotent operators.

We have chosen to give it at the end of the section since it deals with differential equations and thus is not purely linear algebra.

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

Let us now regard D as a linear operator on the subspace V.

This reduction has been accomplished by the general methods of linear algebra, i.e., by the primary decomposition theorem.

that is, one must know something about D other than the fact that it is a linear operator.

Moreover, whereas in Interstate Commerce Commission parlance `` variable cost '' means a cost deemed to vary in direct proportion to changes in rate of output, in the type of analysis now under review `` variable cost '' has been used more broadly, so as to cover costs which, while a function of some one variable ( such as output of energy, or number of customers ), are not necessarily a linear function.

Of the various particle accelerators, the Van De Graff machines, resonant transformers, and linear accelerators are the principal ones available for commercial use.

UTC is a discontinuous time scale composed from segments that are linear transformations of atomic time, the discontinuities being arranged so that UTC approximated UT2 until the end of 1971, and UT1 thereafter.

This was a compromise arrangement for a broadcast time scale: a linear transformation of the BIH's atomic time meant that the time scale was stable and internationally synchronised, while approximating UT1 means that tasks such as navigation which require a source of Universal Time continue to be well served by public time broadcasts.

# As exploratory data analysis, an ANOVA is an organization of an additive data decomposition, and its sums of squares indicate the variance of each component of the decomposition ( or, equivalently, each set of terms of a linear model ).

# Closely related to the ANOVA is a linear model fit with coefficient estimates and standard errors.