The SOLUTION SET of a MATRIX is a point, line, plane or cube

In summary, the dimension of the solution set of a matrix can be determined by finding the dimension of the null-space, which is equal to the rank of the matrix subtracted from the dimension of the matrix. This can be done by using row and column operations to put the matrix in echelon form and counting the number of non-zero rows.
  • #1
math_maj0r
15
0
How do you know if the solution set of a matrix is a point, line, plane or cube? How do you know the dimension of the solution set?

P.S.: This is NOT a homework question. It's a general question about something I'm not 100% clear about
 
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  • #2
The dimension of the solution set is the dimension of the null-space. The dimension of the null-space (nullity) can be found by finding the rank of the matrix (use row and column operations to put the matrix in echelon form and count the number of non-zero rows), and subtracting this from the dimension of the matrix.
 
  • #3
thanks a lot!
 

1. What is the solution set of a matrix?

The solution set of a matrix is the set of all possible solutions to a system of linear equations represented by the matrix.

2. What does it mean for the solution set of a matrix to be a point?

A point solution set means that there is only one unique solution to the system of linear equations represented by the matrix.

3. How is a line solution set represented in a matrix?

A line solution set is represented by a matrix with two variables and two equations, where the equations are linearly dependent.

4. Can the solution set of a matrix be a plane?

Yes, the solution set of a matrix can be a plane if the matrix has three variables and three linearly independent equations. This means that there are infinitely many solutions that lie on a two-dimensional plane.

5. What does a cube solution set in a matrix represent?

A cube solution set is represented by a matrix with three variables and three linearly independent equations. This means that there are infinitely many solutions that lie on a three-dimensional cube.

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