What is the dimension of the solution space of the homogeneous system?

Corollary 8.4. The dimension of the solution space of an n × m homogeneous linear system is m − r where m is the (column) rank of the corresponding coefficient matrix.

What is the dimension of the solution set of the system?

Intuitively, the dimension of a solution set is the number of parameters you need to describe a point in the solution set. For a line only one parameter is needed, and for a plane two parameters are needed.

What can you say the solution space of a linear system if there are more unknowns than equations and at least one solution exists?

If there are more unknowns (n) than the number of equations (m), then the number of solutions of the system is either 0 or ∞. If a system is homogeneous, then it has the zero solution and thus a homogeneous system is always consistent.

What is the dimension of the solution space Ax 0?

Solution : Since A is in row reduced echelon form with 2 leading 1’s the rank of A is 2 and so the solution space of Ax = 0 is 1 dimensional.

What are the properties of a homogeneous system?

1. A homogeneous system is ALWAYS consistent, since the zero solution, aka the trivial solution, is always a solution to that system. 2. A homogeneous system with at least one free variable has infinitely many solutions.

How do you describe a solution set in geometry?

Also, give a geometric description of the solution set. The solution set is a line in 3-space passing thru the point: and parallel to the line that is the solution set of the homogeneous equation. The vectors are linearly dependent if there is more than the trivial solution to the matrix equation .

How do you describe a solution set?

A linear system with a unique solution has a solution set with one element. A linear system with no solution has a solution set that is empty. Solution sets are a challenge to describe only when they contain many elements. …

What is basis of solution?

A basis means each element of the basis is a solution to Ax = 0. Can multiply by a constant and we still get a solution. And we can add together and still get a solution.

What can we say about the number of solutions of an underdetermined system Why?

In general, an underdetermined system of linear equations has an infinite number of solutions, if any. However, in optimization problems that are subject to linear equality constraints, only one of the solutions is relevant, namely the one giving the highest or lowest value of an objective function.

What does it mean to solve a system of linear equations?

If you have a system of equations that contains two equations with the same two unknown variables, then the solution to that system is the ordered pair that makes both equations true at the same time.