A. Duane Porter

The Linear Algebra Curriculum Study Group Recommendations for the First Course in Linear Algebra

There is a growing concern that the linear algebra curriculum at many schools does not adequately address the needs of the students it attempts to serve. In recent years, demand for linear algebra training has risen in client disciplines such as engineering, computer science, operations research, economics, and statis? tics. At the same time, hardware and software improvements in computer science have raised the power of linear algebra to solve problems that are orders of magnitude greater than dreamed possible a few decades ago. Yet in many courses, the importance of linear algebra in applied fields is not communicated to students, and the influence of the computer is not felt in the classroom, in the selection of topics covered or in the mode of presentation. Furthermore, an overemphasis on abstraction may overwhelm beginning students to the point where they leave the course with little understanding or mastery of the basic concepts they may need in later courses and their careers.

Solvability of the Matric Equation AX = B

Abstract Let A be an n×s matrix of rank r, B be an n×t matrix of rank ρ⩽r, and X be an s×t matrix. This paper discusses conditions on the matrices A and B so that the matric equation AX=B will have solutions for the matrix X.

Ranked solutions of AXC=B and AX=B

Abstract Let GF( p n ) denote the finite field of p n elements, p odd. Let A be an s × m matrix of rank ϱ , B be an s × t matrix of rank β, and C be an f × t matrix of rank v . This paper discusses the number of m × f matrices X of rank k over GF( p n ) which are solutions to the matric equations AXC = B or AX = B .

Ranked solutions of some matrix equations

Let GF(pz ) denote the finite field of pz elementsp odd. In this paper formulas are given for calculating the number of ranked solutions X 1,…Xn to the matric equations AX 1… Xn C = B and AX 1…Xn = B as well as the number of ranked solutions U1 ,…Un V 1,…Vm to the matric equation U 1… Un …Un AV 1… Vm = B.

Ranked solutions of the matric equation A1X1=A2X2

Let GF(pz) denote the finite field of pz elements. Let A1 be s×m of rank r1 and A2 be s×n of rank r2 with elements from GF(pz). In this paper, formulas are given for finding the number of X1,X2 over GF(pz) which satisfy the matric equation A1X1=A2X2, where X1 is m×t of rank k1, and X2 is n×t of rank k2. These results are then used to find the number of solutions X1,…,Xn, Y1,…,Ym, m,n>1, of the matric equation A1X1…Xn=A2Y1…Ym.

Partitions of a symmetric matrix over a finite field

Let GF(pw ) denote the finite field with pw elementsp odd. In this paper formulas are given for calculating the number of solutions X 1,…Xn to matrix equations of the form where B is a symmetric matrix. The two cases in which the ranks of the , are specified as arbitrary, but fixed, as well as when the ranks are not specified, are both discussed.

Exponential sums and rectangular partitions

Abstract Let F denote a finite field with q=pf elements, and let σ(A) equal the trace of the square matrix A. This paper evaluates exponential sums of the form S(E,X1,…,Xn)e{−σ(CX1⋯XnE)}, where S(E,X1,…,Xn) denotes a summation over all matrices E,X1,…,Xn of appropriate sizes over F, and C is a fixed matrix. This evaluation is then applied to the problem of counting ranked solutions to matrix equations of the form U1⋯UαA+DV1⋯Vβ=B where A,B,D are fixed matrices over F.

Partitions of a skew matrix over a finite field

Let GF(pw ) denote a finite field with pw elementsp odd. In this paper formulas are given for calculating the number of solutions X1 , … Xn over GF(q) of matrix equations of the form where B is a skew matrix and the prime denotes transpose. The case in which the ranks of the Xv 1  i  n,are specified as arbitrary, but fixed, as well as the case in which the ranks are not specified are both discussed. The number of partitions of certain skew matrices B into a sum of h matrix products, where each product has the form is also discussed.

Systems of bilinear and quadratic equations in a finite field

SummaryAn explicit formula is found for the number of solutions in a finite field of the system of polynomial equations(11), when the coefficients satisfy certain conditions.An explicit formula is found for the number of solutions in a finite field of the system of polynomial equations(11), when the coefficients satisfy certain conditions.