Understanding Homogeneous and Non-Homogeneous Equations in R3

This lesson explores the geometry of solutions for homogeneous and non-homogeneous equations in R3 with two free variables. It details the process of solving these equations, highlighting the differences in their solutions. The homogeneous equation results in a plane through the origin, while the non-homogeneous equation results in a parallel plane shifted by a constant vector. Graphical representations illustrate these concepts, emphasizing the parallel nature of the solution planes.

What is the primary difference between homogeneous and non-homogeneous equations in terms of their solutions?

• Non-homogeneous equations have solutions that form a line.

• Homogeneous equations have solutions that pass through the origin.

• Homogeneous equations have solutions that form a line.

• Non-homogeneous equations have solutions that pass through the origin.

In the homogeneous equation Ax = 0, which variables are considered free?

• x1 and x2

• x1 and x3

• None of the variables

• x2 and x3

How is the solution to the homogeneous equation Ax = 0 parameterized?

• Using real numbers s and t for x1 and x2

• Using real numbers s and t for x2 and x3

• Using real numbers s and t for x1 and x3

• Using real numbers s and t for all variables

What does the graphical representation of the homogeneous solution show?

• A line passing through the origin

• A plane parallel to the x-axis

• A line parallel to the x-axis

• A plane passing through the origin

In the non-homogeneous equation Ax = b, what is the role of the constant vector?

• It has no effect on the solution plane.

• It changes the direction of the solution plane.

• It makes the solution plane pass through the origin.

• It shifts the solution plane away from the origin.

How is the solution to the non-homogeneous equation Ax = b parameterized?

• Using real numbers s and t for all variables

• Using real numbers s and t for x1 and x3

• Using real numbers s and t for x2 and x3

• Using real numbers s and t for x1 and x2

What is the graphical representation of the non-homogeneous solution?

• A line parallel to the homogeneous solution line

• A plane parallel to the homogeneous solution plane

• A plane passing through the origin

• A line passing through the origin

What is the relationship between the planes representing the solutions to the homogeneous and non-homogeneous equations?

• They are perpendicular.

• They are identical.

• They are parallel.

• They intersect at a line.

Why does the non-homogeneous solution plane not pass through the origin?

• It actually does pass through the origin

• Because of the presence of free variables

• Because it is a line, not a plane

• Due to the constant vector shifting the plane

What happens to the homogeneous solution plane when s and t are both zero?

• It becomes a point at the origin.

• It becomes a line.

• It shifts along the x-axis.

• It remains unchanged.