Sentence examples for matrix transformations from from inspiring English sources

Also we will give some matrix transformations from l ( p, λ 2, p ) into l ( q ) by using the matrix given in [4].

In Section 4, we state and prove a general theorem characterizing the matrix transformations from the domain of a triangle matrix to any sequence space.

In this section, we characterize matrix transformations from (b_{c}^{a,b}(B^{(m)})) into (ell_{p}), (ell_{infty}) and c.

In this section, we give two theorems characterizing the classes of matrix transformations from the sequence space f ( G ) into any given sequence space μ and from any sequence space μ into the given sequence space f ( G ).

As an application of this basic theorem, we make a table which gives the necessary and sufficient conditions of the matrix transformations from λ B ( r ˜, s ˜ ) to μ, where λ ∈ { ℓ ∞, c, c 0, ℓ p } and μ ∈ { ℓ ∞, c, c 0, ℓ 1 }.

Here, we shall be concerned with a four-dimensional matrix transformation from any double sequence space λ to any double sequence space μ.

For this reason, in this paragraph, we shall be concerned with matrix transformation from a sequence space X to a sequence space Y.

If x ∈ X implies A x = ( A n ( x ) ) ∈ Y, then we say that A defines a matrix transformation from X into Y, and we denote by ( X, Y ) the class of matrices A which transform X into Y.

If (x=(x_{k})in X) implies that (Axin Y), then we say that A defines a matrix transformation from a sequence space X into another sequence space Y, and we denote the class of such matrices by ((X,Y)).

Finally, we determine the necessary and sufficient conditions on the matrix transformation from the spaces,, and to the spaces and and prove that sequence spaces and have the uniform Opial property for for all.

(Throughout, ∑ k will denote summation over k from k = 1 to k = ∞.) If x = ( x k ) ∈ X implies that A x = ( A n ( x ) ) ∈ Y, we say that A defines a (matrix) transformation from X to Y and we denote it by A : X → Y.