Math 221: LINEAR ALGEBRA Chapter 1. Systems of Linear Equations - PowerPoint PPT Presentation

Math 221: LINEAR ALGEBRA Chapter 1. Systems of Linear Equations §1-3. Homogeneous Equations Le Chen 1 Emory University, 2020 Fall (last updated on 10/21/2020) Creative Commons License (CC BY-NC-SA) 1 Slides are adapted from those by Karen Seyffarth from University of Calgary.

Definition A homogeneous linear equation is one whose constant term is equal to zero. A system of linear equations is called homogeneous if each equation in the system is homogeneous. A homogeneous system has the form  a 11 x 1 + a 12 x 2 + · · · + a 1 n x n = 0    a 21 x 1 + a 22 x 2 + · · · + a 2 n x n = 0  . .  .    a m 1 x 1 + a m 2 x 2 + · · · + a mn x n = 0 where a ij are scalars and x i are variables, 1 ≤ i ≤ m, 1 ≤ j ≤ n.

Definition A homogeneous linear equation is one whose constant term is equal to zero. A system of linear equations is called homogeneous if each equation in the system is homogeneous. A homogeneous system has the form  a 11 x 1 + a 12 x 2 + · · · + a 1 n x n = 0    a 21 x 1 + a 22 x 2 + · · · + a 2 n x n = 0  . .  .    a m 1 x 1 + a m 2 x 2 + · · · + a mn x n = 0 where a ij are scalars and x i are variables, 1 ≤ i ≤ m, 1 ≤ j ≤ n. Remark 1. Notice that x 1 = 0 , x 2 = 0 , · · · , x n = 0 is always a solution to a homogeneous system of equations. We call this the trivial solution. 2. We are interested in finding, if possible, nontrivial solutions (ones with at least one variable not equal to zero) to homogeneous systems.

Example  x 1 + x 2 x 3 + 3 x 4 = 0 −  Solve the system − x 1 + 4 x 2 + 5 x 3 2 x 4 = 0 −  x 1 + 6 x 2 + 3 x 3 + 4 x 4 = 0

Example  x 1 + x 2 x 3 + 3 x 4 = 0 −  Solve the system − x 1 + 4 x 2 + 5 x 3 2 x 4 = 0 −  x 1 + 6 x 2 + 3 x 3 + 4 x 4 = 0 Solution   1 1 − 1 3 0 − 1 4 5 − 2 0   1 6 3 4 0

Example  x 1 + x 2 x 3 + 3 x 4 = 0 −  Solve the system − x 1 + 4 x 2 + 5 x 3 2 x 4 = 0 −  x 1 + 6 x 2 + 3 x 3 + 4 x 4 = 0 Solution     1 1 − 1 3 0 1 0 − 9/5 14/5 0  → · · · → − 1 4 5 − 2 0 0 1 4/5 1/5 0    1 6 3 4 0 0 0 0 0 0

Example  x 1 + x 2 x 3 + 3 x 4 = 0 −  Solve the system − x 1 + 4 x 2 + 5 x 3 2 x 4 = 0 −  x 1 + 6 x 2 + 3 x 3 + 4 x 4 = 0 Solution     1 1 − 1 3 0 1 0 − 9/5 14/5 0  → · · · → − 1 4 5 − 2 0 0 1 4/5 1/5 0    1 6 3 4 0 0 0 0 0 0 The system has infinitely many solutions, and the general solution is  5 s − 14 9 x 1 = 5 t      − 4 5 s − 1 x 2 = 5 t   x 3 = s    x 4 = t

Example  x 1 + x 2 x 3 + 3 x 4 = 0 −  Solve the system − x 1 + 4 x 2 + 5 x 3 2 x 4 = 0 −  x 1 + 6 x 2 + 3 x 3 + 4 x 4 = 0 Solution     1 1 − 1 3 0 1 0 − 9/5 14/5 0  → · · · → − 1 4 5 − 2 0 0 1 4/5 1/5 0    1 6 3 4 0 0 0 0 0 0 The system has infinitely many solutions, and the general solution is    5 s − 14 9 9 5 s − 14 x 1 = 5 t 5 t    x 1       − 4 5 s − 1 − 4 5 s − 1 x 2 = 5 t x 2 5 t     or  = , ∀ s , t ∈ R .     x 3      x 3 = s  s   x 4   x 4 = t t

Theorem If a homogeneous system of linear equations has more variables than equations, then it has a nontrivial solution (in fact, infinitely many).

Definition If X 1 , X 2 , . . . , X p are columns with the same number of entries, and if a 1 , a 2 , . . . a p ∈ R (are scalars) then a 1 X 1 + a 2 X 2 + · · · + a p X p is a linear combination of columns X 1 , X 2 , . . . , X p .

Definition If X 1 , X 2 , . . . , X p are columns with the same number of entries, and if a 1 , a 2 , . . . a p ∈ R (are scalars) then a 1 X 1 + a 2 X 2 + · · · + a p X p is a linear combination of columns X 1 , X 2 , . . . , X p . Example (continued) In the previous example,   5 s − 14 9 5 t   x 1   − 4 5 s − 1 x 2 5 t      =     x 3     s  x 4 t

Definition If X 1 , X 2 , . . . , X p are columns with the same number of entries, and if a 1 , a 2 , . . . a p ∈ R (are scalars) then a 1 X 1 + a 2 X 2 + · · · + a p X p is a linear combination of columns X 1 , X 2 , . . . , X p . Example (continued) In the previous example,   9 5 s − 14 5 t     9 − 14 5 s 5 t   x 1   − 4 5 s − 1  − 4   − 1  x 2 5 t 5 s 5 t          = =  +         x 3    s 0     s  x 4 0 t t

Definition If X 1 , X 2 , . . . , X p are columns with the same number of entries, and if a 1 , a 2 , . . . a p ∈ R (are scalars) then a 1 X 1 + a 2 X 2 + · · · + a p X p is a linear combination of columns X 1 , X 2 , . . . , X p . Example (continued) In the previous example,   9 5 s − 14 5 t     9 − 14 5 s 5 t   x 1   − 4 5 s − 1  − 4   − 1  x 2 5 t 5 s 5 t          = =  +         x 3    s 0     s  x 4 0 t t     9/5 − 14/5 − 4/5 − 1/5     = s  + t     1 0    0 1

Example (continued) This gives us       x 1 9/5 − 14/5 x 2 − 4/5 − 1/5        = s  + t  = sX 1 + tX 2 ,       x 3 1 0    x 4 0 1     9/5 − 14/5 − 4/5 − 1/5     with X 1 = and X 2 =      . 1 0    0 1

Example (continued) This gives us       x 1 9/5 − 14/5 x 2 − 4/5 − 1/5        = s  + t  = sX 1 + tX 2 ,       x 3 1 0    x 4 0 1     9/5 − 14/5 − 4/5 − 1/5     with X 1 = and X 2 =      . 1 0    0 1 The columns X 1 and X 2 are called basic solutions to the original homogeneous system.

Example (continued) Notice that           x 1 9/5 − 14/5 9 − 14 x 2 − 4/5 − 1/5 s − 4  + t − 1            = s  + t =           x 3 1 0 5 5 5 0        x 4 0 1 0 5     9 − 14 − 4 − 1     = r  + q     5 0    0 5 = r (5 X 1 ) + q (5 X 2 ) where r , q ∈ R .

Example (continued)     9 − 14 − 4 − 1     The columns 5 X 1 =  and 5 X 2 =  are also basic solutions     5 0   0 5 to the original homogeneous system.

Example (continued)     9 − 14 − 4 − 1     The columns 5 X 1 =  and 5 X 2 =  are also basic solutions     5 0   0 5 to the original homogeneous system. Remark In general, any nonzero multiple of a basic solution (to a homogeneous system of linear equations) is also a basic solution.

the trivial solution. There is always a solution, and the set of solutions to the system has , there are no parameters, and the system has a unique solution, if many solutions; , there is at least one parameter, and the system has infjnitely if parameters, so What does the rank tell us in the homogeneous case? Suppose A is the augmented matrix of an homogeneous system of m linear equations in n variables, and rank A = r .      0 1 0 ∗ ∗ ∗ ∗ ∗ ∗ ∗    ∗ ∗ ∗ ∗ 0 0 0 1 ∗ 0           ∗ ∗ ∗ ∗ 0 → 0 0 0 1 0     m      ∗ ∗ ∗ ∗ 0 0 0 0 0 0         ∗ ∗ ∗ ∗ 0 0 0 0 0 0 � �� � � �� � n r leading 1 ′ s

the trivial solution. parameters, so , there are no parameters, and the system has a unique solution, if many solutions; , there is at least one parameter, and the system has infjnitely if What does the rank tell us in the homogeneous case? Suppose A is the augmented matrix of an homogeneous system of m linear equations in n variables, and rank A = r .      0 1 0 ∗ ∗ ∗ ∗ ∗ ∗ ∗    ∗ ∗ ∗ ∗ 0 0 0 1 ∗ 0           ∗ ∗ ∗ ∗ 0 → 0 0 0 1 0     m      ∗ ∗ ∗ ∗ 0 0 0 0 0 0         ∗ ∗ ∗ ∗ 0 0 0 0 0 0 � �� � � �� � n r leading 1 ′ s There is always a solution, and the set of solutions to the system has n − r