Some Facts about Linear Systems:
Some not convenient truths:
We educated how to solve a linear system using Mat lab Input the following-
> A = ones(4,4)> b = randn(4,1)> x = A\b
As you will discover there is no solution to the equation Ax = b. This unlucky circumstance is mostly the fault of the matrix A which is also simple its columns (and rows) are all the same. Now try
> b = ones(4,1)> x = [ 1 0 0 0]’> A*x
Therefore the system Ax = b does have a solution. Still unluckily that isn’t the only solution. Try
> x = [ 0 1 0 0]’> A*x
We see that this x is as well a solution Next try > x = [ -4 5 2.27 -2.27]’
This x is a solution! It isn’t hard to see that there are endless possibilities for solutions of this equation.
The largely basic theoretical fact about linear systems is:
Theorem 1 A linear system Ax = b may perhaps have 0, 1 or infinitely many solutions.
Perceptibly in most engineering applications we would want to have exactly one solution. The following two theorems demonstrate that having one and only one solution is a property of A.
Theorem 2 presumes A is a square (n × n) matrix. The subsequent are all equal:
1. The equation Ax = b has precisely one solution for any b.2. det(A) 6= 0.3. A has an inverse.4. The merely solution of Ax = 0 is x = 0.5. The columns of A are linearly independent (as vectors).6. The rows of A are linearly independent.
If A has these properties afterwards it is called non-singular.
Alternatively, a matrix that does not have these properties is called singular.
Theorem 3 presumes A is a square matrix. The following are all equal
1. The equation Ax = b has 0 or ∞ several solutions depending on b.2. det(A) = 0.3. A doesn’t have an inverse.4. The equation Ax = 0 has solutions other than x = 0.5. The columns of A are linearly dependent as vectors.6. The rows of A are linearly dependent.
To observe how the two theorems work, define two matrices (type in A1 then scroll up as well as modify to make A2)
And two vectors:
First compute the determinants of the matrices:
Then attempt to find the inverses:
Which matrix is singular as well as which is non-singular? Ultimately attempt to solve all the possible equations Ax = b:
> x = A1\b1> x = A1\b2> x = A2\b1> x = A2\b2
As you are able to see equations involving the non-singular matrix have one as well as only one solution except equations involving a singular matrix are more complicated.
The residual vector:
Bring to mind that the residual for an approximate solution x of an equation f(x) = 0 is defined as r = f(x).
It is a measure of how nearly the equation is to being satisfied. For a linear system of equations wedescribe the residual of an approximate solution, x by
r = Ax − b. (10.1)
Notice that r is a vector. Its size (norm) is an sign of how shut we have come to solving
Ax = b.
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