A linear system with m equations and n variables is the following set:
a1 1+ a 2 2 ... + n n = b1
m a1 1+ a 2 2 ... + n n = b2
a1 1+ a 2 2 ... + n n = bm
A solution is a vector (1 2t ,.n.,t ) є R that satisfies all the equations.
If the set of solutions is S = Ø then we say that the system is inconsistent.
Note that the above system can be represented as follows:
a11 .12 a |1n 1
a21 .22 a |2n 2
am1 am2 ... mn| bm
Two systems of linear equations are said to be equivalent if and only if they have
the same set of solutions.
Matrix Main diagonal of c is ( 8 4
C = 1 0 6 C є M 3x3
7 8 3 C is a square Matrix order 3
9 6 4 1
C12 0 C 1 ( 1 0 6 ) C = 1
If A є M mxnand m = n then we say that A is a square matrix of order n.
Some special square matrices:
Let A є M nxn
1) A is said to be diagonal if and only if all the elements out of the main
diagonal are zero. The main diagonal CAN have zeroes.
2) A is said to be upper triangular if and only id all its entries below the
main diagonal are zero. Entries above CAN be zero too.
3) A is said to be lower triangular if and only if all its entries above the
main diagonal are zero. Entries below CAN be zero too.
Now we will give an algebraic structure to the space of matrices.
If A,B є M mxn then A = B if and only if they agree entry by entry.
A ijB foijall i,j
If A,B є M mxn then the sum A+B is defined as A+B є M mxn
(A+B) = A + B
ij ij ij
Definition [Scalar Multip