Brief Literature of Linear System

A symmetric matrix is a matrix that gives you the same matrix when transposed.
A skew symmetric matrix is a matrix that gives the negative of the original matrix when transposed.

A column vector is a vector that has just one column.
A row vector is a vector that has one row.

Two matrices are equal if and only if they are of the same size and have the same entries.
Upper triangular matrices are those matrices which has its elements as zero at the main diagonal downwards.
Lower triangular matrices are those matrices which has its elements as zero at the main diagonal upwards.

There are three types of diagonal matrices:

1.   Normal Diagonal Matrix
2.   Scalar Diagonal Matrix
3.   Identity Diagonal Matrix



A homogeneous system is a system where by in the equation Ax = b, all the values of b are equal to zero.

A non homogeneous system is a system where by in the equation Ax = b, at least one of the values of b is not equal to zero.

An underdetermined system is a system where by the number of unknowns in a linear system is greater than the number of equations.
An overdetermined system is a system where by the number of unknowns in a linear system is lesser than the number of equations.
A determined system is a system where by the number of unknowns in a linear system is equal to the number of equations.

A system is said to be consistent if it has at least one solution.
A system is said to be inconsistent if it has no solution at all.

The rank of a matrix A equals the maximum number of linearly independent column vectors A. Hence A and its transpose A has the same rank.

If the determinant of a matrix A is equal to zero, then the matrix is singular.
If the determinant of a matrix A is not equal to zero, then the matrix is non - singular.

If the rank of a matrix equals the rank of the augmented matrix and it is still equal to m. The matrix has a unique solution.

A homogeneous system has a non zero solution. If and only if the determinant of the matrix of coefficient is zero.

A system is controllable if the determinant of S = [B AB A²B ... A^n-1B] is not equal to zero. Else if it is equal to zero, the system is not controllable.

A system is observable if the determinant of V = [D^T A^TD^T (A^T)²D^T ... (A^T)^n-1D^T] is not equal to zero. Else if its is equal to zero, the system is not observable.

There are infinite number of solutions in a system (m x n) where the Rank of A = Rank of A^b = m < n
There is a unique solution in a system where the Rank of A = Rank of A^b = n
There are no solutions exists and the system is inconsistent where Rank of A 
< Rank of A^b

Vector Space

1.   For any vectors U, V, W ∈ V ==>  (U + V) + W = U + (V + W)
2.   There is a vector in V denoted by O and called the zero vector for which U + O = U for any vector U ∈ V

3.   For each vector U ∈ V, there is a vector in V denoted as (-U) for which U + (-U) = 0
4.   For any vector u ∈ V, U + V = V + U
5.   For any scalar k ∈ K and any vector U, V ∈ V ==> k (U + V) = kU + kV
6.   For any scalars a, b ∈ K and any vector u ∈ V for any vector (a + b)U = aU + bU
7.   For any scalars a,b ∈ K  and any vector u ∈ V for any vector (ab)U = a(bU)
8.   For the unit scalar 1 ∈ K, 1u = U for any U ∈ V