Finding the base and dimension of a system of equations

In summary, the given system of equations can be written in matrix form and reduced to a matrix with two identical rows, indicating an infinite number of solutions. The dimension of the solution set of the system of equations is 2. Further row reduction can be done to find the specific solutions.
  • #1
Deimantas
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Homework Statement



Find the base and dimension of a system of equations:

3x1 - 5x2 + 2x3 + 4x4 = 0
7x1 - 4x2 + 1x3 + 3x4 = 0
5x1 + 7x2 - 4x3 - 6x4 = 0

2. The attempt at a solution

Written in matrix form:

3 -5 2 4
7 -4 1 3
5 7 -4 -6

What I get:

11 -3 0 2
7 -4 1 3
11 -3 0 2

so, that's two identical rows. That means there's an infinite number of solutions, I think. What gives? I'm lost. Any help to solving this exercise is appreciated
 
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  • #2
What do you mean you "got" that? How did you get it? Did you do a row reduction? Typically one does a row reduction to get the first column
[tex]\begin{bmatrix}1 \\ 0 \\ 0 \end{bmatrix}[/tex]


Also, you don't mean "find the base and dimension" of the system of equations. You mean to find the dimension of the solution set of the system of equations.
 
  • #3
I'm not very good at this subject, as you can see. Yes, I did row operations on the matrix. I just wanted to know if having two identical rows in a matrix can help in solving this kind of exercise, but I guess not.

Maybe this would look clearer?:

3 -5 2 4 = 3 -5 2 4 = 3 -5 2 4 = 1 -5/3 2/3 4/3
7 -4 1 3 = 21 -12 3 9 = 0 23 -11 -19 = 0 23 -11 -19 = 0 23 -11 -19
5 7 -4 -6 = 15 21 -12 -18 = 0 46 -22 -38 = 0 23 -11 -19 0 23 -11 -19

Is this correct? What next? Like I said, I'm not good at linear algebra :frown:

I can't believe it, the whitespace seems to be ignored in the posts... The answer is in bold, a two rows, 4 columns matrix.
 
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  • #4
Okay, so that reduces to
[tex]\begin{bmatrix}1 & -5 & 2 & 4 \\ 0 & 23 & -11 & -19 \\ 0 & 0 & 0 & 0\end{bmatrix}[/tex]
 

FAQ: Finding the base and dimension of a system of equations

1. How do I find the base of a system of equations?

The base of a system of equations refers to the number of independent equations in the system. To find the base, you can use the rank-nullity theorem, which states that the rank of a matrix (the number of linearly independent rows or columns) plus the nullity of the matrix (the number of free variables) equals the number of columns in the matrix. Therefore, the base of a system of equations is equal to the number of free variables in the system.

2. What is the dimension of a system of equations?

The dimension of a system of equations refers to the number of unknown variables in the system. It is also equal to the number of columns in the coefficient matrix. The dimension can also be determined by finding the number of independent equations in the system, which is the same as the base of the system.

3. Can a system of equations have a base of 0?

No, a system of equations must have at least one independent equation in order to have a base. If the base is 0, it means that the system has no independent equations and therefore has no solution. In other words, the system is inconsistent and cannot be solved.

4. How do I know if a system of equations is consistent or inconsistent?

A system of equations is consistent if it has at least one solution, meaning that the equations can be satisfied simultaneously. It is inconsistent if it has no solution, meaning that the equations contradict each other. To determine consistency, you can use the rank-nullity theorem or solve the system using Gaussian elimination and see if a unique solution is obtained.

5. Can the base and dimension of a system of equations be different?

No, the base and dimension of a system of equations are always the same. This is because the base represents the number of independent equations, while the dimension represents the number of unknown variables. In order for a system of equations to have a solution, the number of independent equations and unknown variables must be equal, hence the base and dimension must also be equal.

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