Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem Proof of the Superposition Principle a n ( x ) y ( n ) 1 ( x )+ ··· + a 1 ( x ) y ′ 1 ( x )+ a 0 ( x ) y 1 ( x ) = 0 a n ( x ) y ( n ) 2 ( x )+ ··· + a 1 ( x ) y ′ 2 ( x )+ a 0 ( x ) y 2 ( x ) = 0 a n ( x ) y ( n ) � � 1 ( x ) + ··· + a 0 ( x ) y 1 ( x ) = c 1 · 0 c 1 a n ( x ) y ( n ) � � + c 2 2 ( x ) + ··· + a 0 ( x ) y 2 ( x ) = c 2 · 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations
Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem Proof of the Superposition Principle a n ( x ) y ( n ) 1 ( x )+ ··· + a 1 ( x ) y ′ 1 ( x )+ a 0 ( x ) y 1 ( x ) = 0 a n ( x ) y ( n ) 2 ( x )+ ··· + a 1 ( x ) y ′ 2 ( x )+ a 0 ( x ) y 2 ( x ) = 0 a n ( x ) y ( n ) � � 1 ( x ) + ··· + a 0 ( x ) y 1 ( x ) = c 1 · 0 c 1 a n ( x ) y ( n ) � � + c 2 2 ( x ) + ··· + a 0 ( x ) y 2 ( x ) = c 2 · 0 a n ( x )( c 1 y 1 ) ( n ) ( x ) + ··· + a 0 ( x )( c 1 y 1 )( x ) = 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations
Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem Proof of the Superposition Principle a n ( x ) y ( n ) 1 ( x )+ ··· + a 1 ( x ) y ′ 1 ( x )+ a 0 ( x ) y 1 ( x ) = 0 a n ( x ) y ( n ) 2 ( x )+ ··· + a 1 ( x ) y ′ 2 ( x )+ a 0 ( x ) y 2 ( x ) = 0 a n ( x ) y ( n ) � � 1 ( x ) + ··· + a 0 ( x ) y 1 ( x ) = c 1 · 0 c 1 a n ( x ) y ( n ) � � + c 2 2 ( x ) + ··· + a 0 ( x ) y 2 ( x ) = c 2 · 0 a n ( x )( c 1 y 1 ) ( n ) ( x ) + ··· + a 0 ( x )( c 1 y 1 )( x ) = 0 a n ( x )( c 2 y 2 ) ( n ) ( x ) � � + + ··· + a 0 ( x )( c 2 y 2 )( x ) = 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations
Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem Proof of the Superposition Principle a n ( x ) y ( n ) 1 ( x )+ ··· + a 1 ( x ) y ′ 1 ( x )+ a 0 ( x ) y 1 ( x ) = 0 a n ( x ) y ( n ) 2 ( x )+ ··· + a 1 ( x ) y ′ 2 ( x )+ a 0 ( x ) y 2 ( x ) = 0 a n ( x ) y ( n ) � � 1 ( x ) + ··· + a 0 ( x ) y 1 ( x ) = c 1 · 0 c 1 a n ( x ) y ( n ) � � + c 2 2 ( x ) + ··· + a 0 ( x ) y 2 ( x ) = c 2 · 0 a n ( x )( c 1 y 1 ) ( n ) ( x ) + ··· + a 0 ( x )( c 1 y 1 )( x ) = 0 a n ( x )( c 2 y 2 ) ( n ) ( x ) � � + + ··· + a 0 ( x )( c 2 y 2 )( x ) = 0 a n ( x )( c 1 y 1 + c 2 y 2 ) ( n ) ( x ) + ··· + a 0 ( x )( c 1 y 1 + c 2 y 2 )( x ) = 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations
Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem Handling Inhomogeneous Equations logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations
Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem Handling Inhomogeneous Equations For the linear inhomogeneous differential equation a n ( x ) y ( n ) ( x )+ a n − 1 ( x ) y ( n − 1 ) ( x )+ ··· + a 1 ( x ) y ′ ( x )+ a 0 ( x ) y ( x ) = g ( x ) let y h ( x ) denote the general solution of the corresponding homogeneous equation. Moreover let y p ( x ) be one particular solution of the inhomogeneous equation. Then the general solution of the inhomogeneous equation is y ( x ) = y p ( x )+ y h ( x ) . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations
Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem Handling Inhomogeneous Equations For the linear inhomogeneous differential equation a n ( x ) y ( n ) ( x )+ a n − 1 ( x ) y ( n − 1 ) ( x )+ ··· + a 1 ( x ) y ′ ( x )+ a 0 ( x ) y ( x ) = g ( x ) let y h ( x ) denote the general solution of the corresponding homogeneous equation. Moreover let y p ( x ) be one particular solution of the inhomogeneous equation. Then the general solution of the inhomogeneous equation is y ( x ) = y p ( x )+ y h ( x ) . So the theory of inhomogeneous equations is pretty much reduced to that of homogeneous equations. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations
Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem Proof logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations
Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem Proof a n ( x ) y ( n ) p ( x ) + ··· + a 0 ( x ) y p ( x ) = g ( x ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations
Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem Proof a n ( x ) y ( n ) p ( x ) + ··· + a 0 ( x ) y p ( x ) = g ( x ) a n ( x ) y ( n ) � � + h ( x ) + ··· + a 0 ( x ) y h ( x ) = 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations
Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem Proof a n ( x ) y ( n ) p ( x ) + ··· + a 0 ( x ) y p ( x ) = g ( x ) a n ( x ) y ( n ) � � + h ( x ) + ··· + a 0 ( x ) y h ( x ) = 0 a n ( x )( y p + y h ) ( n ) ( x ) + ··· + a 0 ( x )( y p + y h )( x ) = g ( x ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations
Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem Proof a n ( x ) y ( n ) p ( x ) + ··· + a 0 ( x ) y p ( x ) = g ( x ) a n ( x ) y ( n ) � � + h ( x ) + ··· + a 0 ( x ) y h ( x ) = 0 a n ( x )( y p + y h ) ( n ) ( x ) + ··· + a 0 ( x )( y p + y h )( x ) = g ( x ) and logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations
Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem Proof a n ( x ) y ( n ) p ( x ) + ··· + a 0 ( x ) y p ( x ) = g ( x ) a n ( x ) y ( n ) � � + h ( x ) + ··· + a 0 ( x ) y h ( x ) = 0 a n ( x )( y p + y h ) ( n ) ( x ) + ··· + a 0 ( x )( y p + y h )( x ) = g ( x ) and a n ( x ) y ( n ) i ( x ) + ··· + a 0 ( x ) y i ( x ) = g ( x ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations
Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem Proof a n ( x ) y ( n ) p ( x ) + ··· + a 0 ( x ) y p ( x ) = g ( x ) a n ( x ) y ( n ) � � + h ( x ) + ··· + a 0 ( x ) y h ( x ) = 0 a n ( x )( y p + y h ) ( n ) ( x ) + ··· + a 0 ( x )( y p + y h )( x ) = g ( x ) and a n ( x ) y ( n ) i ( x ) + ··· + a 0 ( x ) y i ( x ) = g ( x ) a n ( x ) y ( n ) � � − p ( x ) + ··· + a 0 ( x ) y p ( x ) = g ( x ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations
Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem Proof a n ( x ) y ( n ) p ( x ) + ··· + a 0 ( x ) y p ( x ) = g ( x ) a n ( x ) y ( n ) � � + h ( x ) + ··· + a 0 ( x ) y h ( x ) = 0 a n ( x )( y p + y h ) ( n ) ( x ) + ··· + a 0 ( x )( y p + y h )( x ) = g ( x ) and a n ( x ) y ( n ) i ( x ) + ··· + a 0 ( x ) y i ( x ) = g ( x ) a n ( x ) y ( n ) � � − p ( x ) + ··· + a 0 ( x ) y p ( x ) = g ( x ) a n ( x )( y i − y p ) ( n ) ( x ) + ··· + a 0 ( x )( y i − y p )( x ) = 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations
Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem Linear Combinations of Vectors logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations
Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem Linear Combinations of Vectors How do we actually know that several vectors “point in different directions”? logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations
Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem Linear Combinations of Vectors How do we actually know that several vectors “point in different directions”? Let � v 1 ,� v 2 ,...,� v n be vectors. Then any sum n ∑ � v i = c 1 � v 1 + c 2 � v 2 + ··· + c n � c i v n i = 1 with the c i being real numbers is called a linear combination of the vectors. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations
Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem Linear Independence for Vectors logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations
Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem Linear Independence for Vectors A set of n vectors { v 1 , ··· ,� v n } is called linearly dependent if � and only if there are numbers c 1 ,..., c n , which are not all zero, v n = � 0 , where � � v 1 + ··· + c n � such that c 1 0 denotes the null vector , for which all components are zero. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations
Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem Linear Independence for Vectors A set of n vectors { v 1 , ··· ,� v n } is called linearly dependent if � and only if there are numbers c 1 ,..., c n , which are not all zero, v n = � 0 , where � � v 1 + ··· + c n � such that c 1 0 denotes the null vector , for which all components are zero. If no such numbers exist, the set of vectors is called linearly independent . That is, a set of n vectors { v 1 , ··· ,� v n } is called � linearly independent if and only if the only numbers c 1 , ··· , c n , n v i = � ∑ 0 are c 1 = c 2 = ··· = c n = 0. for which c i � i = 1 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations
Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem Determine if the vectors ( 1 , 1 , 3 ) , ( 2 , 4 , 2 ) and ( 3 , − 1 , 4 ) are linearly independent. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations
Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem Determine if the vectors ( 1 , 1 , 3 ) , ( 2 , 4 , 2 ) and ( 3 , − 1 , 4 ) are linearly independent. 1 2 3 0 = + c 2 + c 3 − 1 c 1 1 4 0 3 2 4 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations
Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem Determine if the vectors ( 1 , 1 , 3 ) , ( 2 , 4 , 2 ) and ( 3 , − 1 , 4 ) are linearly independent. 1 2 3 0 = + c 2 + c 3 − 1 c 1 1 4 0 3 2 4 0 + + = 1 c 1 2 c 2 3 c 3 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations
Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem Determine if the vectors ( 1 , 1 , 3 ) , ( 2 , 4 , 2 ) and ( 3 , − 1 , 4 ) are linearly independent. 1 2 3 0 = + c 2 + c 3 − 1 c 1 1 4 0 3 2 4 0 + + = 1 c 1 2 c 2 3 c 3 0 + − = 1 c 1 4 c 2 1 c 3 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations
Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem Determine if the vectors ( 1 , 1 , 3 ) , ( 2 , 4 , 2 ) and ( 3 , − 1 , 4 ) are linearly independent. 1 2 3 0 = + c 2 + c 3 − 1 c 1 1 4 0 3 2 4 0 + + = 1 c 1 2 c 2 3 c 3 0 + − = 1 c 1 4 c 2 1 c 3 0 + + = 3 c 1 2 c 2 4 c 3 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations
Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem Determine if the vectors ( 1 , 1 , 3 ) , ( 2 , 4 , 2 ) and ( 3 , − 1 , 4 ) are linearly independent. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations
Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem Determine if the vectors ( 1 , 1 , 3 ) , ( 2 , 4 , 2 ) and ( 3 , − 1 , 4 ) are linearly independent. + 2 c 2 + 3 c 3 = 0 c 1 + − = c 1 4 c 2 c 3 0 3 c 1 + 2 c 2 + 4 c 3 = 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations
Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem Determine if the vectors ( 1 , 1 , 3 ) , ( 2 , 4 , 2 ) and ( 3 , − 1 , 4 ) are linearly independent. + 2 c 2 + 3 c 3 = 0 c 1 + − = c 1 4 c 2 c 3 0 3 c 1 + 2 c 2 + 4 c 3 = 0 c 1 + 2 c 2 + 3 c 3 = 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations
Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem Determine if the vectors ( 1 , 1 , 3 ) , ( 2 , 4 , 2 ) and ( 3 , − 1 , 4 ) are linearly independent. + 2 c 2 + 3 c 3 = 0 c 1 + − = c 1 4 c 2 c 3 0 3 c 1 + 2 c 2 + 4 c 3 = 0 c 1 + 2 c 2 + 3 c 3 = 0 − = 2 c 2 4 c 3 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations
Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem Determine if the vectors ( 1 , 1 , 3 ) , ( 2 , 4 , 2 ) and ( 3 , − 1 , 4 ) are linearly independent. + 2 c 2 + 3 c 3 = 0 c 1 + − = c 1 4 c 2 c 3 0 3 c 1 + 2 c 2 + 4 c 3 = 0 c 1 + 2 c 2 + 3 c 3 = 0 − = 2 c 2 4 c 3 0 − − = 4 c 2 5 c 3 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations
Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem Determine if the vectors ( 1 , 1 , 3 ) , ( 2 , 4 , 2 ) and ( 3 , − 1 , 4 ) are linearly independent. + 2 c 2 + 3 c 3 = 0 c 1 + − = c 1 4 c 2 c 3 0 3 c 1 + 2 c 2 + 4 c 3 = 0 c 1 + 2 c 2 + 3 c 3 = 0 − = 2 c 2 4 c 3 0 − − = 4 c 2 5 c 3 0 + + = c 1 2 c 2 3 c 3 0 − = 2 c 2 4 c 3 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations
Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem Determine if the vectors ( 1 , 1 , 3 ) , ( 2 , 4 , 2 ) and ( 3 , − 1 , 4 ) are linearly independent. + 2 c 2 + 3 c 3 = 0 c 1 + − = c 1 4 c 2 c 3 0 3 c 1 + 2 c 2 + 4 c 3 = 0 c 1 + 2 c 2 + 3 c 3 = 0 − = 2 c 2 4 c 3 0 − − = 4 c 2 5 c 3 0 + + = c 1 2 c 2 3 c 3 0 − = 2 c 2 4 c 3 0 − = 13 c 3 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations
Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem Determine if the vectors ( 1 , 1 , 3 ) , ( 2 , 4 , 2 ) and ( 3 , − 1 , 4 ) are linearly independent. + 2 c 2 + 3 c 3 = 0 c 1 + − = c 1 4 c 2 c 3 0 3 c 1 + 2 c 2 + 4 c 3 = 0 c 1 + 2 c 2 + 3 c 3 = 0 − = 2 c 2 4 c 3 0 − − = 4 c 2 5 c 3 0 + + = c 1 2 c 2 3 c 3 0 − = 2 c 2 4 c 3 0 − = 13 c 3 0 0 = c 3 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations
Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem Determine if the vectors ( 1 , 1 , 3 ) , ( 2 , 4 , 2 ) and ( 3 , − 1 , 4 ) are linearly independent. + 2 c 2 + 3 c 3 = 0 c 1 + − = c 1 4 c 2 c 3 0 3 c 1 + 2 c 2 + 4 c 3 = 0 c 1 + 2 c 2 + 3 c 3 = 0 − = 2 c 2 4 c 3 0 − − = 4 c 2 5 c 3 0 + + = c 1 2 c 2 3 c 3 0 − = 2 c 2 4 c 3 0 − = 13 c 3 0 0 = c 3 = c 2 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations
Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem Determine if the vectors ( 1 , 1 , 3 ) , ( 2 , 4 , 2 ) and ( 3 , − 1 , 4 ) are linearly independent. + 2 c 2 + 3 c 3 = 0 c 1 + − = c 1 4 c 2 c 3 0 3 c 1 + 2 c 2 + 4 c 3 = 0 c 1 + 2 c 2 + 3 c 3 = 0 − = 2 c 2 4 c 3 0 − − = 4 c 2 5 c 3 0 + + = c 1 2 c 2 3 c 3 0 − = 2 c 2 4 c 3 0 − = 13 c 3 0 0 = c 3 = c 2 = c 1 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations
Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem Determine if the vectors ( 1 , 1 , 3 ) , ( 2 , 4 , 2 ) and ( 3 , − 1 , 4 ) are linearly independent. + 2 c 2 + 3 c 3 = 0 c 1 + − = c 1 4 c 2 c 3 0 3 c 1 + 2 c 2 + 4 c 3 = 0 c 1 + 2 c 2 + 3 c 3 = 0 − = 2 c 2 4 c 3 0 − − = 4 c 2 5 c 3 0 + + = c 1 2 c 2 3 c 3 0 − = 2 c 2 4 c 3 0 − = 13 c 3 0 0 = c 3 = c 2 = c 1 , and the vectors are linearly independent. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations
Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem − 1 2 3 and are − 2 − 2 Determine if , 2 − 4 − 5 3 linearly independent. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations
Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem − 1 2 3 and are − 2 − 2 Determine if , 2 − 4 − 5 3 linearly independent. − + = 2 c 1 1 c 2 3 c 3 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations
Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem − 1 2 3 and are − 2 − 2 Determine if , 2 − 4 − 5 3 linearly independent. − + = 2 c 1 1 c 2 3 c 3 0 − 2 c 1 + − = 2 c 2 2 c 3 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations
Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem − 1 2 3 and are − 2 − 2 Determine if , 2 − 4 − 5 3 linearly independent. − + = 2 c 1 1 c 2 3 c 3 0 − 2 c 1 + − = 2 c 2 2 c 3 0 − 4 c 1 + − = 3 c 2 5 c 3 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations
Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem − + = 2 c 1 c 2 3 c 3 0 − 2 c 1 + 2 c 2 − 2 c 3 = 0 − 4 c 1 + − = 3 c 2 5 c 3 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations
Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem − + = 2 c 1 c 2 3 c 3 0 − 2 c 1 + 2 c 2 − 2 c 3 = 0 − 4 c 1 + − = 3 c 2 5 c 3 0 − + = 2 c 1 c 2 3 c 3 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations
Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem − + = 2 c 1 c 2 3 c 3 0 − 2 c 1 + 2 c 2 − 2 c 3 = 0 − 4 c 1 + − = 3 c 2 5 c 3 0 − + = 2 c 1 c 2 3 c 3 0 c 2 + c 3 = 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations
Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem − + = 2 c 1 c 2 3 c 3 0 − 2 c 1 + 2 c 2 − 2 c 3 = 0 − 4 c 1 + − = 3 c 2 5 c 3 0 − + = 2 c 1 c 2 3 c 3 0 c 2 + c 3 = 0 + = c 2 c 3 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations
Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem − + = 2 c 1 c 2 3 c 3 0 − 2 c 1 + 2 c 2 − 2 c 3 = 0 − 4 c 1 + − = 3 c 2 5 c 3 0 − + = 2 c 1 c 2 3 c 3 0 c 2 + c 3 = 0 + = c 2 c 3 0 c 2 = − c 3 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations
Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem − + = 2 c 1 c 2 3 c 3 0 − 2 c 1 + 2 c 2 − 2 c 3 = 0 − 4 c 1 + − = 3 c 2 5 c 3 0 − + = 2 c 1 c 2 3 c 3 0 c 2 + c 3 = 0 + = c 2 c 3 0 c 2 = − c 3 , c 1 = c 2 − 3 c 3 2 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations
Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem − + = 2 c 1 c 2 3 c 3 0 − 2 c 1 + 2 c 2 − 2 c 3 = 0 − 4 c 1 + − = 3 c 2 5 c 3 0 − + = 2 c 1 c 2 3 c 3 0 c 2 + c 3 = 0 + = c 2 c 3 0 c 2 = − c 3 , c 1 = c 2 − 3 c 3 , 2 choose c 3 = 1 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations
Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem − + = 2 c 1 c 2 3 c 3 0 − 2 c 1 + 2 c 2 − 2 c 3 = 0 − 4 c 1 + − = 3 c 2 5 c 3 0 − + = 2 c 1 c 2 3 c 3 0 c 2 + c 3 = 0 + = c 2 c 3 0 c 2 = − c 3 , c 1 = c 2 − 3 c 3 , 2 choose c 3 = 1: c 2 = − 1 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations
Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem − + = 2 c 1 c 2 3 c 3 0 − 2 c 1 + 2 c 2 − 2 c 3 = 0 − 4 c 1 + − = 3 c 2 5 c 3 0 − + = 2 c 1 c 2 3 c 3 0 c 2 + c 3 = 0 + = c 2 c 3 0 c 2 = − c 3 , c 1 = c 2 − 3 c 3 , 2 choose c 3 = 1: c 2 = − 1, c 1 = − 2. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations
Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem − + = 2 c 1 c 2 3 c 3 0 − 2 c 1 + 2 c 2 − 2 c 3 = 0 − 4 c 1 + − = 3 c 2 5 c 3 0 − + = 2 c 1 c 2 3 c 3 0 c 2 + c 3 = 0 + = c 2 c 3 0 c 2 = − c 3 , c 1 = c 2 − 3 c 3 , 2 choose c 3 = 1: c 2 = − 1, c 1 = − 2. 2 − 1 3 0 = − 2 − 2 − + − 2 2 0 − 4 3 − 5 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations
Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem − + = 2 c 1 c 2 3 c 3 0 − 2 c 1 + 2 c 2 − 2 c 3 = 0 − 4 c 1 + − = 3 c 2 5 c 3 0 − + = 2 c 1 c 2 3 c 3 0 c 2 + c 3 = 0 + = c 2 c 3 0 c 2 = − c 3 , c 1 = c 2 − 3 c 3 , 2 choose c 3 = 1: c 2 = − 1, c 1 = − 2. 2 − 1 3 0 = , − 2 − 2 − + − 2 2 0 − 4 3 − 5 0 and the vectors are linearly dependent. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations
Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem Why use Matrices? logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations
Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem Why use Matrices? 1 1 3 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations
Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem Why use Matrices? 1 2 1 4 3 2 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations
Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem Why use Matrices? 1 2 3 − 1 1 4 3 2 4 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations
Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem Why use Matrices? 1 2 3 − 1 1 4 3 2 4 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations
Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem Matrices logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations
Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem Matrices Let m and n be positive integers. An m × n -matrix is a rectangular array of mn numbers a ij , commonly indexed and written as follows. ··· a 11 a 12 a 1 ( n − 1 ) a 1 n ··· a 21 a 22 a 2 ( n − 1 ) a 2 n ··· a 31 a 32 a 3 ( n − 1 ) a 3 n A =( a i , j ) i = 1 ,..., m = . . . . . . j = 1 ,..., n a ( m − 1 ) 1 a ( m − 1 ) 2 ··· a ( m − 1 )( n − 1 ) a ( m − 1 ) n a m 1 a m 2 ··· a m ( n − 1 ) a mn The index i is called the row index and the index j is called the column index . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations
Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem Determinants logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations
Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem Determinants � a 11 � a 12 Let A = be a 2 × 2 matrix. Then we define the a 21 a 22 determinant of A to be � �� � a 11 � � a 11 a 12 a 12 � � det ( A ) : = det : = � : = a 11 a 22 − a 12 a 21 . � � a 21 a 22 a 21 a 22 � logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations
Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem Determinants � a 11 � a 12 Let A = be a 2 × 2 matrix. Then we define the a 21 a 22 determinant of A to be � �� � a 11 � � a 11 a 12 a 12 � � det ( A ) : = det : = � : = a 11 a 22 − a 12 a 21 . � � a 21 a 22 a 21 a 22 � Let A = ( a ij ) i , j = 1 ,..., n be a square matrix and let A ij be the matrix obtained by erasing the i th row and the j th column. Then the determinant of A is defined recursively by n n ( − 1 ) i + j a ij det ( A ij ) = ( − 1 ) i + j a ij det ( A ij ) , ∑ ∑ det ( A ) : = | A | : = j = 1 i = 1 where the i in the first sum is an arbitrary row and the j in the second sum is an arbitrary column. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations
Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem Uses of the Determinant logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations
Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem Uses of the Determinant 1. The determinant gives the n -dimensional volume of the parallelepiped spanned by the column vectors. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations
Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem Uses of the Determinant 1. The determinant gives the n -dimensional volume of the parallelepiped spanned by the column vectors. 2. The n -dimensional vectors � v 1 ,...,� v n are linearly independent if and only if det ( � v 1 ,...,� v n ) � = 0 , where ( v n ) denotes a matrix whose columns are the � v 1 ,...,� vectors � v i . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations
Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem Uses of the Determinant 1. The determinant gives the n -dimensional volume of the parallelepiped spanned by the column vectors. 2. The n -dimensional vectors � v 1 ,...,� v n are linearly independent if and only if det ( � v 1 ,...,� v n ) � = 0 , where ( v n ) denotes a matrix whose columns are the � v 1 ,...,� vectors � v i . 3. Computation of characteristic polynomials. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations
Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem 1 2 3 and are linearly − 1 Determine if 1 , 4 3 2 4 independent. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations
Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem 1 2 3 and are linearly − 1 Determine if 1 , 4 3 2 4 independent. 1 2 3 − 1 det 1 4 3 2 4 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations
Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem 1 2 3 and are linearly − 1 Determine if 1 , 4 3 2 4 independent. 1 2 3 − 1 det 1 4 3 2 4 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations
Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem 1 2 3 and are linearly − 1 Determine if 1 , 4 3 2 4 independent. 1 2 3 − 1 det 1 4 3 2 4 � 4 � − 1 = 1 · det 2 4 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations
Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem 1 2 3 and are linearly − 1 Determine if 1 , 4 3 2 4 independent. 1 2 3 − 1 det 1 4 3 2 4 � 4 � − 1 = 1 · det 2 4 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations
Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem 1 2 3 and are linearly − 1 Determine if 1 , 4 3 2 4 independent. 1 2 3 − 1 det 1 4 3 2 4 � 4 � − 1 = 1 · det 2 4 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations
Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem 1 2 3 and are linearly − 1 Determine if 1 , 4 3 2 4 independent. 1 2 3 − 1 det 1 4 3 2 4 � 4 � � 2 � − 1 3 = 1 · det − 1 · det 2 4 2 4 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations
Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem 1 2 3 and are linearly − 1 Determine if 1 , 4 3 2 4 independent. 1 2 3 − 1 det 1 4 3 2 4 � 4 � � 2 � − 1 3 = 1 · det − 1 · det 2 4 2 4 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations
Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem 1 2 3 and are linearly − 1 Determine if 1 , 4 3 2 4 independent. 1 2 3 − 1 det 1 4 3 2 4 � 4 � � 2 � − 1 3 = 1 · det − 1 · det 2 4 2 4 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations
Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem 1 2 3 and are linearly − 1 Determine if 1 , 4 3 2 4 independent. 1 2 3 − 1 det 1 4 3 2 4 � 4 � � 2 � � 2 � − 1 3 3 = 1 · det − 1 · det + 3 · det 2 4 2 4 4 − 1 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations
Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem 1 2 3 and are linearly − 1 Determine if 1 , 4 3 2 4 independent. 1 2 3 − 1 det 1 4 3 2 4 � 4 � � 2 � � 2 � − 1 3 3 = 1 · det − 1 · det + 3 · det 2 4 2 4 4 − 1 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations
Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem 1 2 3 and are linearly − 1 Determine if 1 , 4 3 2 4 independent. 1 2 3 − 1 det 1 4 3 2 4 � 4 � � 2 � � 2 � − 1 3 3 = 1 · det − 1 · det + 3 · det 2 4 2 4 4 − 1 = 1 · 18 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations
Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem 1 2 3 and are linearly − 1 Determine if 1 , 4 3 2 4 independent. 1 2 3 − 1 det 1 4 3 2 4 � 4 � � 2 � � 2 � − 1 3 3 = 1 · det − 1 · det + 3 · det 2 4 2 4 4 − 1 = 1 · 18 − 1 · 2 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations
Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem 1 2 3 and are linearly − 1 Determine if 1 , 4 3 2 4 independent. 1 2 3 − 1 det 1 4 3 2 4 � 4 � � 2 � � 2 � − 1 3 3 = 1 · det − 1 · det + 3 · det 2 4 2 4 4 − 1 = 1 · 18 − 1 · 2 + 3 · ( − 14 ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations
Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem 1 2 3 and are linearly − 1 Determine if 1 , 4 3 2 4 independent. 1 2 3 − 1 det 1 4 3 2 4 � 4 � � 2 � � 2 � − 1 3 3 = 1 · det − 1 · det + 3 · det 2 4 2 4 4 − 1 = 1 · 18 − 1 · 2 + 3 · ( − 14 ) = − 26 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations
Existence and Uniqueness Linear Independence Matrices and Determinants Linear Independence Revisited Solution Theorem 1 2 3 and are linearly − 1 Determine if 1 , 4 3 2 4 independent. 1 2 3 − 1 det 1 4 3 2 4 � 4 � � 2 � � 2 � − 1 3 3 = 1 · det − 1 · det + 3 · det 2 4 2 4 4 − 1 = 1 · 18 − 1 · 2 + 3 · ( − 14 ) = − 26 � = 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Theory of Linear Ordinary Differential Equations
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