System of Linear Algebraic Functions
December 8, 2016
Categorised in: 1st Year Maths 1
Condition for consistency of non-homogenous equation
Consider, AX = B
m = total no. of equations, n = toal no. of unknowns
Case: m ≠ n
ρ(A) = ρ(A, B) i.e. system is consistent.
ρ(A) ≠ ρ(A, B) i.e. system is inconsistent and hence no solution
ρ(A) = ρ(A, B) = n = total no. of unknowns, hence unique solution i.e. only one solution
ρ(A) = ρ(A, B) = r < n, hence infinite solutions which can be represented parametrically by employing some parameter t.
Thus, r unknowns can be expressed in terms of remaining (n-r) unknowns.
Case: m = n > 3
ρ(A) = ρ(A, B) = n, system is consistent and possesses unique solution.
ρ(A) ≠ ρ(A, B), system is inconsistent and possesses no solution.
ρ(A) = ρ(A, B) = r < n, system is consistent and possesses infinite solutions
Case: m = n = 3
Find |A|
|A| ≠ 0, system is consistent, $A^{-1}$ exists and possesses unique solution given by X = $A^{-1}$ B
|A| = 0, apply method of reduction and write augmented matrix (A, B).
Now, ρ(A) = ρ(A, B) < n then system possesses an infinite no. of solutions.
if ρ(A) ≠ ρ(A, B) then system is inconsistent and possesses no solution.
Vector Dependancies:
Linear dependent:
$x_1,x_2,…,x_n$ are vectors
$c_1,c_2,…,c_n$ are n scalars not all zero
Thus, $c_1 x_1 + c_2 x_2 +…+ c_n x_n = 0$
Linear independent:
$x_1,x_2,…,x_n$ are vectors
$c_1,c_2,…,c_n$ are n scalars not all zero
Thus, $c_1 x_1 + c_2 x_2 +…+ c_n x_n = 0$ should imply $c_1 = c_2 …. c_n = θ$
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