Page 64 - 35Linear Algebra

64 Systems of Linear Equations

2. No solutions

3. Infinitely many solutions

2.5.1 The Geometry of Solution Sets: Hyperplanes

Consider the following algebra problems and their solutions.

1. 6x = 12 has one solution: 2.

2a. 0x = 12 has no solution.

2b. 0x = 0 has infinitely many solutions; its solution set is R.

In each case the linear operator is a 1 × 1 matrix. In the first case, the linear
operator is invertible. In the other two cases it is not. In the first case, the
solution set is a point on the number line, in case 2b the solution set is the
whole number line.
Lets examine similar situations with larger matrices: 2 × 2 matrices.

6 0 x 12 2
1. = has one solution: .
0 2 y 6 3

1 3 x 4
2a. = has no solutions.
0 0 y 1

1 3 x 4 4 −3
2bi. = has solution set + y : y ∈ R .
0 0 y 0 0 1

0 0 x 0 x
2bii. = has solution set : x, y ∈ R .
0 0 y 0 y
Again, in the first case the linear operator is invertible while in the other
cases it is not. When a 2 × 2 matrix from a matrix equation is not invertible
the solution set can be empty, a line in the plane, or the plane itself.
For a system of equations with r equations and k veriables, one can have a
number of different outcomes. For example, consider the case of r equations
in three variables. Each of these equations is the equation of a plane in three-
dimensional space. To find solutions to the system of equations, we look for
the common intersection of the planes (if an intersection exists). Here we
have five different possibilities:

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