Lecture 4 The Gauß scheme A linear system of equations Matrix algebra deals essentially with linear linear systems. Multiplicative elements. A non-linear system Solving simple stoichiometric equations

The division through a vector or a matrix is not defined! 2 equations and four unknowns Solving a linear system

Det A: determinant of A The determinant of linear dependent matrices is zero. Such matrices are called singular. Determinants

Higher order determinants for any i =1 to n The matrix is linear dependent The number of operations raises with the faculty of n. Laplace formula

For a non-singular square matrix the inverse is defined as r 2 =2r 1 r 3 =2r 1 +r 2 Singular matrices are those where some rows or columns can be expressed by a linear combination of others. Such columns or rows do not contain additional information. They are redundant. A linear combination of vectors A matrix is singular if it’s determinant is zero. Det A: determinant of A A matrix is singular if at least one of the parameters k is not zero.

The augmented matrix The trace of a square matrix is the sum of its diagonal entries. An insect species at three locations has the following abundances per season The predation rates per season are given by The diagonal entries (trace) of the dot product of AB’ contain the total numbers of insects per site kept by predators

(AB) -1 = B -1 A -1 ≠ A -1 B -1 Determinant The inverse of a 2x2 matrixThe inverse of a diagonal matrix The inverse of a square matrix only exists if its determinant differs from zero. Singular matrices do not have an inverse The inverse can be unequivocally calculated by the Gauss-Jordan algorithm

Systems of linear equations Determinant

Solving a simple linear system

Identity matrix Only possible if A is not singular. If A is singular the system has no solution. The general solution of a linear system Systems with a unique solution The number of independent equations equals the number of unknowns. X: Not singularThe augmented matrix X aug is not singular and has the same rank as X. The rank of a matrix is minimum number of rows/columns of the largest non-singular submatrix

A matrix is linear independent if none of the row or column vectors can be expressed by a linear combinations of the remaining vectors r 2 =2r 1 r 3 =2r 1 +r 2 The matrices are linear dependent A linear combination of vectors A matrix of n-vectors (row or columns) is called linear dependent if it is possible to express one of the vectors by a linear combination of the other n-1 vectors. If a vector V of a matrix is linear dependent on the other vecors, V does not contain additional information. It is completely defined by the other vectors. The vector V is redundant. Linear independence

How to detect linear dependency Any solution of k 3 =0 and k 1 =-2k 2 satisfies the above equations. The matrix is linear dependent. The rank of a matrix is the maximum number of linearly independent row and column vectors If a matrix A is linearly independent, then any submatrix of A is also linearly independent

Consistent Rank(A) = rank(A:B) = n Consistent Rank(A) = rank(A:B) < n Inconsistent Rank(A) < rank(A:B) Consistent Rank(A) = rank(A:B) < n Inconsistent Rank(A) < rank(A:B) Consistent Rank(A) = rank(A:B) = n Infinite number of solutions No solution Infinite number of solutions No solution Infinite number of solutions

We have only four equations but five unknowns. The system is underdetermined. The missing value is found by dividing the vector through its smallest values to find the smallest solution for natural numbers.

Equality of atoms involved Including information on the valences of elements We have 16 unknows but without experminetnal information only 11 equations. Such a system is underdefined. A system with n unknowns needs at least n independent and non-contradictory equations for a unique solution. If n i and a i are unknowns we have a non-linear situation. We either determine n i or a i or mixed variables such that no multiplications occur.

The matrix is singular because a 1, a 7, and a 10 do not contain new information Matrix algebra helps to determine what information is needed for an unequivocal information. From the knowledge of the salts we get n 1 to n 5

We have six variables and six equations that are not contradictory and contain different information. The matrix is therefore not singular.

Linear models in biology tN The logistic model of population growth K denotes the maximum possible density under resource limitation, the carrying capacity. r denotes the intrinsic population growth rate. If r > 1 the population growths, at r < 1 the population shrinks. We need four measurements

N t K Overshot We have an overshot. In the next time step the population should decrease below the carrying capacity. Population growth tN NN K/2 Fastest population growth