Solutions Sets of Linear Systems

1.5 Solutions Sets of Linear Systems Math 2331 – Linear Algebra 1.5 Solutions Sets of Linear Systems Math 2331, Linear Algebra 1 / 12 1.5 Solutions Sets of Linear Systems Homogeneous Nonhomogeneous 1.5 Solutions Sets of Linear Systems Homogeneous System Nontrivial Solutions Writing Solution Set in Parametric Vector Form Nonhomogeneous System Theorem: Solutions of Nonhomogeneous System Writing Solution Set in Parametric Vector Form Math 2331, Linear Algebra 2 / 12 1.5 Solutions Sets of Linear Systems Homogeneous Nonhomogeneous Homogeneous System Homogeneous System Ax = 0 (A is m × n and 0 is the zero vector in Rm ) Example x1 + 10x2 = 0 2x1 + 20x2 = 0 Corresponding matrix equation Ax = 0:      1 10 x1 0 = x2 2 20 0   0 Trivial solution: x = or x=0 0 Math 2331, Linear Algebra 3 / 12 1.5 Solutions Sets of Linear Systems Homogeneous Nonhomogeneous Homogeneous System: Nontrivial Solutions The homogeneous system Ax = 0 always has the trivial solution, x = 0. Nontrivial Solution Nonzero vector solutions are called nontrivial solutions. Example (cont.) Do nontrivial solutions exist?     1 10 0 1 10 0 ∼ 2 20 0 0 0 0 Consistent system with a free variable has infinitely many solutions. A homogeneous equation Ax = 0 has nontrivial solutions if and only if the system of equations has . Math 2331, Linear Algebra 4 / 12 1.5 Solutions Sets of Linear Systems Homogeneous Nonhomogeneous Homogeneous System: Example 1 Example (1) Determine if the following homogeneous system has nontrivial solutions and then describe the solution set. 2x1 + 4x2 − 6x3 = 0 4x1 + 8x2 − 10x3 = 0 Solution: There is at least one free variable (why?) =⇒ nontrivial solutions exist       2 4 −6 0 1 2 −3 0 1 2 −3 0 ∼ ∼ 4 8 −10 0 4 8 −10 0 0 0 2 0 x1 =  ∼ 1 2 0 0 0 0 1 0  =⇒ x2 is free x3 = Math 2331, Linear Algebra 5 / 12 1.5 Solutions Sets of Linear Systems Homogeneous Nonhomogeneous Homogeneous System: Example 1 (cont.)     x1 −2x2 x =  x2  =  x2  = x3 0   −2  1  = x2 v 0 Graphical representation: solution set = span{v} = line through 0 in R3 Math 2331, Linear Algebra 6 / 12 1.5 Solutions Sets of Linear Systems Homogeneous Nonhomogeneous Nonhomogeneous System: Example 2 Example (2) Describe the solution set of 2x1 + 4x2 − 6x3 = 0 4x1 + 8x2 − 10x3 = 4 ( same left side as in the previous example) Solution:   2 4 −6 0 row reduces to 4 8 −10 4   x1 x =  x2  = x3 Math 2331, Linear Algebra  1 2 0 6 0 0 1 2  7 / 12 1.5 Solutions Sets of Linear Systems Homogeneous Nonhomogeneous Nonhomogeneous System: Example 2 (cont.)     6 −2 x =  0  + x2  1  = p+x2 v 2 0 Graphical representation: Parallel solution sets of Ax = 0 & Ax = b Math 2331, Linear Algebra 8 / 12 1.5 Solutions Sets of Linear Systems Homogeneous Nonhomogeneous Nonhomogeneous System: Recap of Previous Two Examples Example (1. Solution of Ax = 0)   −2 x = x2  1  = x2 v 0 x = x2 v = parametric equation of line passing through 0 and v Example (2. Solution of Ax = b)     6 −2 x =  0  + x2  1  = p+x2 v 2 0 x = p+x2 v = parametric equation of line passing through p parallel to v Math 2331, Linear Algebra 9 / 12 1.5 Solutions Sets of Linear Systems Homogeneous Nonhomogeneous Nonhomogeneous System: Theorem Parallel solution sets of Ax = b and Ax = 0 Theorem Suppose the equation Ax = b is consistent for some given b, and let p be a solution. Then the solution set of Ax = b is the set of all vectors of the form w = p + vh , where vh is any solution of the homogeneous equation Ax = 0. Math 2331, Linear Algebra 10 / 12 1.5 Solutions Sets of Linear Systems Homogeneous Nonhomogeneous Nonhomogeneous System: Example Example Describe the solution set of 2x1 − 4x2 − 4x3 = 0; compare it to the solution set 2x1 − 4x2 − 4x3 = 6. Solution: Corresponding augmented matrix to 2x1 − 4x2 − 4x3 = 0:   2 −4 −4 0 ∼ (fill-in) Vector form of the solution:   2x2 + 2x3 = v =  x2 x3   2  1 + 0   2  0  1 Corresponding augmented matrix to 2x1 − 4x2 − 4x3 = 6:   2 −4 −4 6 ∼ (fill -in) Math 2331, Linear Algebra 11 / 12 1.5 Solutions Sets of Linear Systems Homogeneous Nonhomogeneous Nonhomogeneous System: Example (cont.) Vector form of the solution:    3 + 2x2 + 2x3 = x2 v = x3  +   2  1 + 0   2  0  1 Parallel Solution Sets of Ax = 0 and Ax = b Math 2331, Linear Algebra 12 / 12