Linear System Solution in Matrix Form

In summary: You have found the correct solution. Now, here is the summary of the conversation:In summary, the given system of equations has two equations and five unknowns, meaning that up to two unknowns can be calculated while the other three remain unknown. The solution space is a three-dimensional subspace of R5, and the solution can be written in terms of three vectors in a matrix form. The solution space is called the kernel or nullspace of the transformation represented by the matrix A. The solution found by the expert and the one given in the book are the same, but there may be other solutions by isolating different variables and finding different linear combinations of the three vectors.
  • #1
0kelvin
50
5
\begin{cases}
x+ 2y - z + w - t = 0 \\
x - y + z + 3w - 2t = 0
\end{cases}

Add 1st to the 2nd:

$$2x + y + w - t = 0 \\
y = -2x -w + t = 0$$

Substitute y in the 1st:

##x + 2x + w - t + 3w - 2t + z = 0 \\
z = 3x - 4w + 3t##

Both z and y in terms of x,w,t. Writing using matrix form:

##\left[\begin{matrix}
x \\
y \\
z \\
w \\
t \\
\end{matrix}\right] =
x\left[\begin{matrix}
1 \\
-2 \\
-3 \\
0 \\
0 \\
\end{matrix}\right] +
w\left[\begin{matrix}
0 \\
-1 \\
-4 \\
1 \\
0 \\
\end{matrix}\right] +
t\left[\begin{matrix}
0 \\
1 \\
3 \\
0 \\
1 \\
\end{matrix}\right]
##

Solution ##x(1,-2,-3,0,0) + w(0,-1,-4,1,0) + t(0,1,3,0,1)##. I can isolate any two variables, but I have no idea if different solutions are in fact, part of the same set of solutions.
 
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  • #2
How many unknowns do you have? How many equations? What does that tell you about the solution?
 
  • #3
Two equations of five unknowns, that means that up to two unknowns can be calculated, while the other three remain unknown.

the answer in the book is also a sum of three vectors, in terms of x,w and t. But my values aren't equal nor multiples.
 
Last edited:
  • #4
0kelvin said:
Two equations of five unknowns, that means that up to two unknowns can be calculated, while the other three remain unknown.

the answer in the book is also a sum of three vectors, in terms of x,w and t. But my values aren't equal nor multiples.
You can check to see if your solution is actually a solution this way.
Write your system of equations in matrix form:
$$\begin{bmatrix} 1 & 2 & -1 & 1 & -1 \\ 1 & -1 & 1 & 3 & -2\end{bmatrix}$$
Multiply each of your three vectors on the left by this matrix. If you get the zero vector for each multiplication, your vectors are correct.

You can check to see whether the vectors given as answers in your book are correct by doing the same thing.

I haven't checked your work, but from what you have, the solution space is a three-dimensional subspace of R5. Your vectors, if correct, are a basis for that subspace. It's possible that the vectors given in your textbook are also correct. If so, they just happen to be a different set of basis vectors.

BTW, I don't know if you're covered this yet, but the solution space you have found is called the kernel or nullspace of the tranformation represented by the matrix I showed. If we call this matrix A, the kernel is the set of vectors x such that Ax = 0.
 
  • #5
My solution is wrong. The first vector multiplied by the matrix results in 0, but not the other two. The answer in the book is correct because all three vectors multiplied by the matrix result in 0.

There was a mistake in the first addition. Fixed it and now I've found the same answer as in the book. This means that there are other solutions, I can isolate any two variables and find different linear combinatinos of three vectors.
 
Last edited:
  • #6
Well done.
 

What is a linear system?

A linear system is a set of equations that can be represented graphically as a set of intersecting lines, and the solution to the system is the point where all the lines intersect.

How do I solve a linear system?

To solve a linear system, you can use various methods such as substitution, elimination, or graphing. These methods involve manipulating the equations to find the values of the variables that satisfy all the equations in the system.

What is the importance of solving linear systems?

Solving linear systems is essential in many fields of science, including physics, engineering, and economics. It allows us to find the relationships between different variables and make predictions or solve real-world problems.

What are the different types of linear systems?

There are two types of linear systems: consistent and inconsistent. A consistent system has at least one solution, while an inconsistent system has no solution. A consistent system can also be further classified as independent, dependent, or overdetermined.

Can a linear system have more than one solution?

Yes, a linear system can have infinitely many solutions. This occurs when the equations are equivalent, meaning they represent the same line, and all points on that line are solutions to the system.

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