Some facts about Linear Systems in Linear Algebra

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]’

> A*x

This x is a solution! It isn’t hard to see that there are endless possibilities for solutions of this equation.

Basic theory:

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)

1275_matrices.jpg

And two vectors:

2295_vectors.jpg

First compute the determinants of the matrices:

>det(A1)
>det(A2)

Then attempt to find the inverses:

>inv(A1)
>inv(A2)

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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