Help with Gaussian Elimination

In summary, Gaussian Elimination is a mathematical method that efficiently solves systems of linear equations by using elementary row operations to reduce the equations to a simpler form. It is useful in various areas of mathematics and science, but has limitations such as only being applicable to linear equations and becoming more complex with larger systems.
  • #1
jtm
19
0
S =

Columns 1 through 3

1.0000 0 0
0 1.0000 0
0 0 1.0000

Columns 4 through 5

0.2750 -0.2786
-0.1750 0.5929
0.2250 0.1357

Which is the rref of

A =

2 9 9 1 6
2 7 3 0 4
9 6 7 3 2

(i) Which columns of S are the pivot columns?
(ii) Which variables xi are the free variables?
(iii) What is the rank of R?
(iv) What is the rank of A?
(v) What is the nullity of A?
(vi) Why does the equation Ax = b have a solution?


I put my answers as:
i Columns 1 2 3 are pivot columns in S.
ii x4 and x5 are free variables because we are in R^3.
iii Rank of R is 3.
iv Rank of A is 3 also.
v Nullity of A is 0.
vi Ax = b has a solution because in the rref of [A b] is consistant, and since it was consistant we had 3 variables (x1-x3) (because of R^3 space) which depend on the free variables and 2 free variables (x4-x5).

I am having trouble with mostly vi. Just because this matrix in rref is supposed to have x4 and x5 in R^3 space? I'm lost. Help! Thanks!
 
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  • #2
Since no one seems to want to touch this you're going to have to settle for an explanation from a B teamer like me.

i) Okay.
ii) True, x4 and x5 are free but it's not because you're "in" R^3. A 3 x 5 matrix could have pivot columns of 1, 4 and 5, which would make x2 and x3 free.
iii) Okay
iv) Okay
v) This is wrong. Check your definition of Nullity.
vi) [A b] is either consistent or inconsistent. If and only if it's inconsistent the matrix RREF(A b) has a special property. What is that property? Based on what we know of RREF(A) ... can that property every be satisfied? Hint: No it can't.
 
Last edited:
  • #3


I would like to clarify and provide a more detailed explanation for the answers given above.

i) The pivot columns in S are columns 1, 2, and 3. These are the columns that contain the leading entries (or pivots) in the rref of S. These pivots determine the basic variables in the system.

ii) In this case, the free variables are x4 and x5. These are the variables that do not have a pivot in their respective columns in the rref of S. They are also known as the non-basic variables.

iii) The rank of R is 3 because it has 3 pivot columns. The rank of a matrix is equal to the number of pivot columns in its rref.

iv) The rank of A is also 3 because it is equal to the number of pivot columns in its rref. This means that the columns of A are linearly independent and span a 3-dimensional space.

v) The nullity of A is 0 because it is equal to the number of free variables in the system. Since there are no free variables, the null space (or set of solutions to Ax = 0) is only the trivial solution.

vi) The equation Ax = b has a solution because the rref of [A b] is consistent. This means that there exists at least one solution to the system of equations. In this case, there are 3 basic variables (x1-x3) and 2 free variables (x4-x5) which determine the values of the basic variables. The solution to the system is given by [x1, x2, x3, x4, x5] = [a, b, c, d, e], where a, b, and c are determined by the pivots in the rref of [A b] and d and e can be chosen arbitrarily as they correspond to the free variables. This is possible because the columns of A are linearly independent and span a 3-dimensional space, allowing for a unique solution to the system.
 

1. What is Gaussian Elimination?

Gaussian Elimination is a mathematical method used to solve systems of linear equations. It involves manipulating the equations through a series of steps to reduce them to an equivalent system with fewer unknown variables. This process is used to find the values of the variables that satisfy all the equations in the system.

2. Why is Gaussian Elimination useful?

Gaussian Elimination is useful because it allows us to solve systems of linear equations efficiently and systematically. It is also used in many other areas of mathematics and science, such as in matrix operations and solving differential equations.

3. How does Gaussian Elimination work?

Gaussian Elimination works by using elementary row operations to manipulate the equations in a system. These operations include multiplying an equation by a constant, swapping two equations, and adding a multiple of one equation to another. By performing these operations in a specific order, the equations are reduced to a simpler form that can be solved more easily.

4. What are the steps involved in Gaussian Elimination?

The steps involved in Gaussian Elimination include setting up the system of linear equations in augmented matrix form, performing elementary row operations to reduce the matrix to echelon form, and then using back substitution to solve for the variables.

5. Are there any limitations to Gaussian Elimination?

Yes, there are some limitations to Gaussian Elimination. It can only be used to solve systems of linear equations, so it cannot be applied to nonlinear systems. It also may not work if the equations are dependent or inconsistent. Additionally, as the size of the system increases, the calculations involved in Gaussian Elimination become more complex and time-consuming.

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