Intuitive interpretation of the solution sets of AX = B and AX = 0

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For an intuitive interpretation of the solution sets of AX = B and AX = 0 consider the case of one or more linearly independent equations in three unknowns and their physical interpretations in three-dimensional space.

Consider the following cases:

Case 1. The linear system AX = B consists of just one linear equation in three unknowns. This equation would correspond to a plane in three-dimensional space. The points on this plane would constitute a solution set for the system AX = B. Every point on the plane is a solution to the system and every solution to the system lies on the plane. The matrix A would have dimensions 1x3 and have a rank of 1.

Case 2. The system AX = B consists of two linearly independent equations in three unknowns. These two equations would be represented by two planes in three-dimensional space and the solution set for the system would correspond to the points on the line of intersection of the two planes. The matrix A would be of dimensions 2x3 and have a rank of 2.

Case 3. The system AX = B consists of three linearly independent equations in three unknowns. This would be represented in three-dimensional space as three planes and the solution set for the system would consist of the single point of intersection of those three planes. The matrix would have dimensions 3x3 and have a rank of 3.

Case 4. The system AX = B consists of m linear equations in three unknowns in which only two of the equations are independent and all the rest are dependent. This would be represented in three-dimensional space as a pencil of m planes all passing through a common line of intersection. The solution set for the system would be all the points on the common line of intersection of the m planes. The matrix A would have dimensions mx3 and have a rank of 2.

Now consider the system AX = 0. Here the physical interpretation is the same as with the system AX = B except that in the system AX = 0 all the planes pass through the origin of the coordinate system. Thus in the case of a single linear equation in three unknowns we would have a single plane passing through the origin and the solution set of the system would correspond to all points on that plane. For the case of two linearly independent equations in three unknowns we would have two planes passing through the origin and the solution set would consist of all points on the line of intersection of those two planes. For the case of three linearly independent equations in three unknowns we would have three planes passing through the origin and the solution set of the system would correspond to that single point of intersection of those three planes, the point (0,0,0), the origin of the coordinate system. For the case of m linear equations in three unknowns in which only two of the equations are independent and all of the rest are dependent we would have a pencil of planes all passing through the origin and about some common line of intersection and the solution set of the system would correspond to all of the points on that common line of intersection of the planes.

With all of this said we are now in a position to provide a physical interpretation in three-dimensional space for the following theorem regarding a complete solution of the linear system AX = B.

Theorem. Let Ax = b be a consistent system having n unknowns, and let the rank of A be r.

Case 1. r = n. There is a single solution vector x.

Case 2. r < n. Let x_p be a particular solution to the system Ax = b. Vector x_p can be any vector that satisfies the system. Let u₁, u₂, .... ,u_n-r be any set of linearly independent vectors that span the solution space of the system Ax = 0. There are n-r such vectors. In other words, one can find n-r linearly independent vectors u₁, u₂, .... ,u_n-r which satisfy the set of homogeneous equations Ax = 0. The vector x_p plus any linear combination of these vectors u₁, u₂, .... ,u_n-r is a solution of the given equation. There are no other solutions. If b = 0, the vector x_p can be taken as x_p = 0. The complete solution can thus be written as

X = x_p + c₁u₁ + c₂u₂ + .... + c_n-ru_n-r

where c₁, c₂, .... ,c_n-r are arbitrary constants.

The complete solution of AX = B then consists of a particular solution vector X plus any linear combination of u₁, u₂, .... ,u_n-r. If B = 0 the vector X of the particular solution can be taken as X = 0.

Let us try to visualize the physical meaning of the theorem by some simple examples in three-dimensional space.

Suppose the system AX = B consists of the single equation

5x + 3y + 9z = 13.

This corresponds to a plane in three-dimensional space. Envision this plane in space and label it "K". This plane represents the solution set of the system. Pass a plane through the origin parallel to plane K and label it "L". Plane L represents the solution set to the homogeneous system AX = 0, which consists of the single equation

5x + 3y + 9z = 0

Label the origin of the coordinate system "O" and pick any point on plane K and label it "P". Vector OP represents a particular solution to the system. Label any two points on plane L as "M" and "N". Vectors OM and ON represent two basis vectors for plane L. The complete solution of the system is then given by OP + a OM + b ON where a and b are arbitrary numbers. Note that the sum of these three vectors represents a point on plane K i.e. it represents some point in the solution set of AX = B.

Suppose the system AX = B consists of the two equations

2x + 5y + 3z = 8

9x - 2y + 8z = 20

This corresponds to two planes in three-dimensional space that intersect in some line. Label the line of their intersection "K". This line represents the solution set of the system. Pass a line through the origin of the coordinate system parallel to line K and label it "L". Line L represents the solution set to the homogeneous system AX = 0 which is the system

2x + 5y + 3z = 0

9x - 2y + 8z = 0

Label the origin of the coordinate system "O" and label any point on line K as "P". Vector OP represents a particular solution to the system. Label some point on line L as "M". Vector OM represents a basis vector for line L. The complete solution of the system is then given by OP + a OM where a is any arbitrary number. The sum of these two vectors represents some point on line K and corresponds to some point in the solution set of AX = B.