Math 211 Math 211 Lecture #21 Determinants October 16, 2002 2 - PowerPoint PPT Presentation

1 Math 211 Math 211 Lecture #21 Determinants October 16, 2002

2 Basis of a Subspace Basis of a Subspace A set of vectors v 1 , v 2 , . . . , and v k form a Definition: basis of a subspace V if 1. V = span( v 1 , v 2 , . . . , v k ) 2. v 1 , v 2 , . . . , and v k are linearly independent. • The best way to describe a subspace is to give a basis. Return Span

3 Examples of Bases Examples of Bases • The vector v = (1 , − 1 , 1) T is a basis for null( A ) . � null( A ) is the subspace of R 3 with basis v . • The vectors v = (1 , − 1 , 1 , 0) T and w = (0 , − 2 , 0 , 1) T form a basis for null( B ) . � null( B ) is the subspace of R 4 with basis { v , w } . Return

4 Existence of a Basis Existence of a Basis Let V be a subspace of R n . Proposition: 1. If V � = { 0 } , then V has a basis. 2. Bases are not unique, but every basis of V has the same number of elements. The dimension of a subspace V is the Definition: number of elements in a basis of V . Return Examples

5 Another Example of a Nullspace Another Example of a Nullspace 3 − 3 1 − 1 1 − 1 0 0     − 2 2 − 1 1 0 0 1 − 1 rref     A = − →     1 − 1 0 0 0 0 0 0     13 − 13 5 − 5 0 0 0 0 • null( A ) is the subspace of R 4 with basis (1 , 1 , 0 , 0) T and (0 , 0 , 1 , 1) T . • null( A ) has dimension 2. • In M ATLAB , use commands null(A) or null(A,’r’) . Return

6 Nonsingular Matrices Nonsingular Matrices Let A be an n × n matrix. We know the following: • A is nonsingular if the equation A x = b has a solution for any right hand side b . (This is the definition.) • If A is nonsingular then A x = b has a unique solution for any right hand side b . • A is singular if and only if the homogeneous equation A x = 0 has a non-zero solution. � null( A ) is non-trivial ⇔ A is singular. Return Theorem

7 Determinants in 2D Determinants in 2D • How do we decide if a matrix A is nonsingular? • A is nonsingular if and only if when put into row echelon form, the matrix has nonzero entries along the diagonal. • Example: the general 2 × 2 matrix � a � b A = c d is nonsingular if and only if ad − bc � = 0 . � We define ad − bc to be the determinant of A . Return

8 Determinants in 3D Determinants in 3D a 11 a 12 a 13   A = a 21 a 22 a 23   a 31 a 32 a 33 • The same (but more difficult) argument shows that A is nonsingular if and only if a 11 a 22 a 33 − a 11 a 23 a 32 − a 12 a 21 a 33 + a 12 a 23 a 31 − a 13 a 22 a 31 + a 13 a 21 a 32 � = 0 . • This will be the determinant of A . Return 2D

9 Main Theorem Main Theorem We will define the determinant of a square matrix A so that the next theorem is true. The n × n matrix A is nonsingular if and Theorem: only if det( A ) � = 0 . If A is an n × n matrix, then null( A ) Corollary: contains a nonzero vector if and only if det( A ) = 0 . • The corollary contains the most important fact about determinants for ODEs. Return

10 Matrices and Minors Matrices and Minors The general n × n matrix has the form · · · a 11 a 12 a 1 n   · · · a 21 a 22 a 2 n   A = . . .   ... . . .   . . .   · · · a n 1 a n 2 a nn The ij -minor of an n × n matrix A is the Definition: ( n − 1) × ( n − 1) matrix A ij obtained from A by deleting the i th row and the j th column. Return

11 Definition of Determinant Definition of Determinant The determinant of an n × n matrix A is Definition: defined to be n � ( − 1) j +1 a 1 j det( A 1 j ) . det( A ) = j =1 • The definition is inductive. � It assumes we know how to compute the determinants of ( n − 1) × ( n − 1) matrices. � We start with the 2 × 2 matrix. 3 × 3 Return Matrix

12 Example Example 2 1 0   det 3 − 2 4   − 1 5 3 � − 2 4 � = ( − 1) 2 × 2 × det 5 3 � 3 4 � + ( − 1) 3 × 1 × det − 1 3 = 2 × ( − 26) − 13 = − 65 Definition

13 Expansion by the i th Row Expansion by the i th Row For any i , we have n � ( − 1) i + j a ij det( A ij ) . det( A ) = j =1 • This is called expansion by the i th row. • Example: 5 − 6 3 � 5   3 �  = 4 · det det 0 4 0 = 156 .  2 9 2 − 16 9 Return Definition

14 Properties of the Determinant Properties of the Determinant • The formula for the determinant of a matrix A is the sum of n ! products of the entries of A (sometimes × − 1 .) � Each summand is the product of n entries, one from each row, and one from each column. • The determinant of a triangular matrix is the product of the diagonal terms. � We can use row operations to compute determinants. Return

15 Row Operations and Determinants Row Operations and Determinants If B is obtained from A by • adding a multiple of one row to another, then det( B ) = det( A ) . • interchanging two rows, then det( B ) = − det( A ) . • multiplying a row by c � = 0 , then det( B ) = c det( A ) . Return

16 Example Example − 5 2 3   A = 25 − 9 − 12   10 7 17 det( A ) = 50 Return Row operations

17 More Properties More Properties • If A has two equal rows , then det( A ) = 0 . • If A has a row of all zeros , then det( A ) = 0 . • det( A T ) = det( A ) . • If A has two equal columns, then det( A ) = 0 . • If A has a column of all zeros, then det( A ) = 0 . Return

18 Column Operations and Determinants Column Operations and Determinants If B is obtained from A by • adding a multiple of one column to another, then det( B ) = det( A ) . • interchanging two columns, then det( B ) = − det( A ) . • multiplying a column by c � = 0 , then det( B ) = c det( A ) . Return

19 Expansion by a Column Expansion by a Column We can also expand by a column. n � ( − 1) i + j a ij det( A ij ) . det( A ) = i =1 • This is called expansion by the j th column. det( A T ) = det( A ) Return Expansion by row

20 Example Example − 5 − 6 0   A = 3 4 0   − 8 − 16 9 � − 5 − 6 � det( A ) = 9 · det 3 4 . = 9 · ( − 2) = − 18 Return Expansion by row Expansion by column

21 Determinants and Bases Determinants and Bases A collection of n vectors v 1 , v 2 , . . . , v n Proposition: in R n is a basis for R n if and only if det([ v 1 v 2 . . . v n ]) � = 0 . Theorem

22 Examples Examples 1 1 1 1   2 1 − 1 − 2   det  = 1 .   − 2 − 1 1 1  2 2 1 1 3 − 1 0 1   12 − 6 0 5   det  = − 1 .   32 − 15 − 3 13  18 − 10 − 1 8 Expansion by row Expansion by column Row operations Column operations

23 The Span of a Set of Vectors The Span of a Set of Vectors The span of a set of vectors is the set of all Definition: linear combinations of those vectors. The span of the vectors v 1 , v 2 , . . . , and v k is denoted by span( v 1 , v 2 , . . . , v k ) . Return

24 Linear Independence Linear Independence The vectors v 1 , v 2 , . . . , and v k are linearly Definition: independent if the only linear combination of them which is equal to the zero vector is the one with all of the coefficients equal to 0. • In symbols, c 1 v 1 + c 2 v 2 + · · · + c k v k = 0 ⇒ c 1 = c 2 = · · · = c k = 0 . Return

25 Example of a Nullspace Example of a Nullspace 4 3 − 1 1 0 − 1     rref A = − 3 − 2 1 − → 0 1 1     1 2 1 0 0 0 The nullspace of A is the set null( A ) = { a v | a ∈ R } , where v = (1 , − 1 , 1) T . • The nullspace of A consists of all multiples of v . Return

26 Another Example of a Nullspace Another Example of a Nullspace 4 3 − 1 6 1 0 − 1 0     rref B = − 3 − 2 1 − 4 − → 0 1 1 2     1 2 1 4 0 0 0 0 The nullspace of B is the set null( B ) = { a v + b w | a, b ∈ R } , where v = (1 , − 1 , 1 , 0) T and w = (0 , − 2 , 0 , 1) T . • null( B ) consists of all linear combinations of v and w . Return