Let $X$ and $Y$ be vector spaces over the same field $K$. Following the terminology of Richard Arens [Pacific J. Math. 11 (1961), 9–23], a relation $F$ of $X$ into $Y$ is called linear if $\lambda F\left(x\right)\subset F\left(\lambda x\right)$ and $F\left(x\right)+F\left(y\right)\subset F(x+y)$ for all $\lambda \in K\setminus \left\{0\right\}$ and $x,y\in X$. After improving and supplementing some former results on linear relations, we show that a relation $\Phi$ of a linearly independent subset $E$ of $X$ into $Y$ can be extended to a linear relation $F$ of $X$ into $Y$ if and only if there exists a linear subspace $Z$ of $Y$ such that $\Phi \left(e\right)\in Y|Z$ for all $e\in E$. Moreover, if $E$ generates...

The maximal column rank of an m by n matrix is the maximal number of the columns of A which are linearly independent. We compare the maximal column rank with rank of matrices over a nonbinary Boolean algebra. We also characterize the linear operators which preserve the maximal column ranks of matrices over nonbinary Boolean algebra.

The Boolean rank of a nonzero $m\times n$ Boolean matrix $A$ is the minimum number $k$ such that there exist an $m\times k$ Boolean matrix $B$ and a $k\times n$ Boolean matrix $C$ such that $A=BC$. In the previous research L. B. Beasley and N. J. Pullman obtained that a linear operator preserves Boolean rank if and only if it preserves Boolean ranks $1$ and $2$. In this paper we extend this characterizations of linear operators that preserve the Boolean ranks of Boolean matrices. That is, we obtain that a linear operator preserves Boolean rank...

Let $A$ be a Boolean $\{0,1\}$ matrix. The isolation number of $A$ is the maximum number of ones in $A$ such that no two are in any row or any column (that is they are independent), and no two are in a $2\times 2$ submatrix of all ones. The isolation number of $A$ is a lower bound on the Boolean rank of $A$. A linear operator on the set of $m\times n$ Boolean matrices is a mapping which is additive and maps the zero matrix, $O$, to itself. A mapping strongly preserves a set, $S$, if it maps the set $S$ into the set $S$ and the complement of...

Let ${𝐌}_{m,n}$ be the set of all $m\times n$ real matrices. A matrix $A\in {𝐌}_{m,n}$ is said to be row-dense if there are no zeros between two nonzero entries for every row of this matrix. We find the structure of linear functions $T:{𝐌}_{m,n}\to {𝐌}_{m,n}$ that preserve or strongly preserve row-dense matrices, i.e., $T\left(A\right)$ is row-dense whenever $A$ is row-dense or $T\left(A\right)$ is row-dense if and only if $A$ is row-dense, respectively. Similarly, a matrix $A\in {𝐌}_{n,m}$ is called a column-dense matrix if every column of $A$ is a column-dense vector. At the end, the structure of linear...

We introduce linear transformations of Euclidean topological spaces given by a transformation matrix. Next, we prove selected properties and basic arithmetic operations on these linear transformations. Finally, we show that a linear transformation given by an invertible matrix is a homeomorphism.

We prove a number of theorems concerning various notions used in the theory of continuity of barycentric coordinates.

We consider a commutative ring $R$ with identity and a positive integer $N$. We characterize all the 3-tuples $(L_1,L_2,L_3)$ of linear transforms over $R^N$, having the “circular convolution” property, i.eṡuch that $x*y=L_3(L_1\left(x\right)\otimes L_2\left(y\right))$ for all $x,y\in R^N$.