Systems of equations revisited, Mathematics

Assignment Help:

Systems of Equations Revisited

We require doing a quick revisit of systems of equations. Let's establish with a general system of equations.

a11 x1 + a12 x2 +................+a1n xn = b1

a21 x1 + a22 x2 +.............. +a2n xn  = b2

...................

an1 x1 + an2 x2 +............... +ann xn  = bn                              ...................(1)

Here, covert each side in a vector to find,

1410_Systems of Equations Revisited.png

The left side of such equation can be thought of like a matrix multiplication.

845_Systems of Equations Revisited1.png

Simplifying up the notation a little provides,

A x? = b?   ................................(2)

Now there, x? is a vector that elements are the unknowns in the original system of equations.

We take (2) the matrix form of the system of equations (1) and resolving (2) is equal to solving

(1). the solving process is identical. The augmented matrix for (2) is,

A(b?)

Once we contain the augmented matrix we proceed as we did along with a system which hasn't been wrote in matrix form.

We also have the subsequent fact regarding to solutions to (2).

Fact

Provided the system of equation (2) we contain one of the subsequent three possibilities for solutions.

1.   There will be no more solutions.

2.   There will be particularly one solution.

3.   There will be infinitely various solutions.

Actually we can go a little farther here. Because we are assuming that we've got similar number of equations like unknowns the matrix A in (2) is a square matrix and therefore we can calculate its determinant. This provides the following fact.

Fact

Provided the system of equations in (2) we have the subsequent.

1.   If A is nonsingular then there will be particularly one solution to the system.

2.   If A is singular then there will either be no solution or infinitely various solutions to the system.

The matrix form of a homogeneous system is as,

A x?= 0?

 Here 0? is the vector of all zeroes. Under the homogeneous system we are guaranteed to have a solutions, x? = 0?. The fact above for homogeneous systems is so, Fact

Given the homogeneous system (3) we contain the subsequent.

1.   If A is nonsingular then the only solution will be x? = 0?

2.   If A is singular then there will be infinitely many nonzero solutions to the system.


Related Discussions:- Systems of equations revisited

Simplify the boolean function, Simplify the Boolean function: F...

Simplify the Boolean function: F (w,x,y,z) = ∑ (0, 1, 2, 3, 4, 6, 8, 9, 12, 13, 14)  (8)  Ans:   f(w, x, y, z) = ∑(0, 1, 2, 3, 4, 6, 8, 9, 12, 13, 14) The above

Sketch the graph f ( x ) = - x5 + (5/2 )x4 + (40/3) x3 + 5, Sketch the grap...

Sketch the graph of the below function. f ( x ) = - x 5 + (5/2 )x 4 + (40/3) x 3 + 5 Solution : Whenever we sketch a graph it's good to have a few points on the graph to

Which of the subsequent decimals is the greatest number, Which of the subse...

Which of the subsequent decimals is the greatest number? If you add zeros to the end of every of the numbers so that each number has 5 places after the decimal point, it is sim

Cubic math, A fish tank has the base area of 45 cm3 and is filled to the de...

A fish tank has the base area of 45 cm3 and is filled to the depth of 12 cm.If the height is 25 cm then how much more will be needed to fill the rest of the tank?

Distance is given then find the value of k, In the graphical representatio...

In the graphical representation of a frequency distribution if the distance between mode and mean is k times the distance between median and mean then find the value of k.

Arc length - applications of integrals, Arc Length - Applications of integr...

Arc Length - Applications of integrals In this part we are going to look at determining the arc length of a function.  As it's sufficiently easy to derive the formulas that we'

Determine the area of the sail, If a triangular sail has a horizontal lengt...

If a triangular sail has a horizontal length of 30 ft and a vertical height of 83 ft , Determine the area of the sail? a. 1,245 ft 2 b. 1,155 ft 2 c. 201 ft 2 d. 2,4

the system by graphing, Suppose you are in the market for a new home and a...

Suppose you are in the market for a new home and are interested in a new housing community under construction in a another city. a) The sales representative later shows that there

Tables and funcuctions, write an equation for a functionthat gives the valu...

write an equation for a functionthat gives the value in ech table .

Write Your Message!

Captcha
Free Assignment Quote

Assured A++ Grade

Get guaranteed satisfaction & time on delivery in every assignment order you paid with us! We ensure premium quality solution document along with free turntin report!

All rights reserved! Copyrights ©2019-2020 ExpertsMind IT Educational Pvt Ltd