# Solving Line Space Problem with Matrices A & B

• Cyannaca
In summary, the conversation discusses finding the value of x in two matrices, A and B, so that they have the same row space. The speaker is unsure of the definition of "line space" and suggests it may mean "row space". They are instructed to row reduce the matrices and find the span, which is equivalent to finding solutions to simultaneous equations. The use of reduced row echelon form can make the answer easier to read.
Cyannaca
Consider the matrices

A=
(1 5 -3)
(2 x -3)

B=
(1 3 -1)
(3 x -1)

Find x so A and B have the same line space.
I tried to do this problem but I was stuck with

A=
(1 (x-5) 0)
(0 (x-10) 3)

B=
(1 3 -1)
(0 (x-9) 2)

Also, I'm not really sure of what they mean by same line space.

If you don't know what the definition of "line space" is from your notes, then we're not guaranteed to know. Indeed the only definition I could find was that it was a Liouville space, which is a product of Hilbert spaces, so I don't think that's waht you want.

Is it possible they mean "row" space? Or "column" space (unlikely, since they have the same column space in $\mathbb{R}^2$ regardless of the value of x)? Ask your intructor.

Yes, they mean row space. Sorry, I translated it from French a bit too fast I guess. Actually, I have to find the value of x so they have the same row space but I don't know how.

Row reduce into ecehlon form, the resulting vectors must have the same spane. Finding the span is the same as finding solutions to simultaneous equations. If you use reduced row echelon form the answer is even easier to read off. These topics will almost certainly bi in your notes, though I don't know the French terms for them (though I can guess what echelon is).

## 1. What is the "Line Space Problem"?

The Line Space Problem is a mathematical concept that involves finding all possible linear combinations of a set of vectors in a vector space. This problem is often solved using matrices and is essential in fields such as linear algebra and computer graphics.

## 2. How are matrices A & B used to solve the Line Space Problem?

Matrices A & B are used to represent the set of vectors in the Line Space Problem. Matrix A represents the coefficients of the vectors, while matrix B represents the actual vectors. By using matrix operations, we can find the linear combinations of the vectors and determine the line space.

## 3. What is the significance of solving the Line Space Problem?

Solving the Line Space Problem is crucial in various mathematical and scientific fields such as linear algebra, computer graphics, and physics. It helps us understand the concept of vector spaces and their properties, which are essential in analyzing and solving complex problems.

## 4. Can matrices A & B be used to solve other problems besides the Line Space Problem?

Yes, matrices A & B have many applications in various fields, such as solving systems of linear equations, finding eigenvalues and eigenvectors, and performing transformations in computer graphics.

## 5. Are there any limitations to using matrices A & B to solve the Line Space Problem?

While matrices A & B are powerful tools for solving the Line Space Problem, they do have limitations. For example, the size of the matrices may become too large to handle, or the problem may not have a unique solution. In such cases, alternative methods may need to be used.

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