MATH 6 LINEAR ALGEBRA LECTURE NOTES FALL - These Lecture Notes are not in a final form being still subject of improvement
Contents Systems of linear equations and matrices 5 Introduction to systems of linear equations 5 Gaussian elimination 7 3 Matrices and matrix operations 4 Inverses Rules of matrix arithmetic 8 5 Elementary matrices and a method for finding A 6 Further results on systems of equations and invertibility 5 7 Diagonal, Triangular and Symmetric Matrices 7 Determinants 33 The determinant function 33 Evaluating determinants by row reduction 35 3 Properties of determinant function 36 4 Cofactor expansion Cramer s rule 38 3 Euclidean vector spaces 43 3 Euclidean n-space 43 3 Linear transformations from R n to R m 46 33 Properties of linear transformations from R n to R m 5 3
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Chapter Systems of linear equations and matrices Introduction to systems of linear equations Definition A linear equation in the n variables x,, x n, equally called unknowns, is the problem of finding the values of x,, x n such that a x + + a n x n b, where a,, a n, b are constants A solution of a linear equation a x + + a n x n b is a sequence t,, t n of n real numbers such that the equation is satisfied when we replace x t,, x n t n The set of all solutions of the equation is called its solution set or the general solution Definition A finite set of linear equations in the variables x,, x n linear equations or a a linear system A solution of a linear system is called a system of a x + a x + + a n x n b a x + a x + + a n x n b () a m x + a m x + + a mn x n b m with m linear equations and n unknowns, is a sequence t,, t n of n real numbers such that each equation of the system is satisfied when we replace x t,, x n t n The set of all solutions of the linear system is called its solution set or the general solution A system of equations is said to be consistent if it has a solution at least Otherwise it is called inconsistent The augmented matrix 5
of the linear system (3) is a a a n b a a a n b a m a m a mn b m Remark 3 The solution set of a linear system unchanges if we perform on the system one of the following operations: Multiply an equation with a nonzero constant; Interchange two equations; 3 Add a multiple of one equation to another The corresponding operations at the level of augmented matrix are: Multiply a row with a nonzero constant; Interchange two rows; 3 Add a multiple of one row to another Example 4 Solve the following linear system x + y 3z + aw 4 3x y + 5z + aw 4x + y + z + aw Solution: The augmented matrix of the system is 3 a 3 5 a 4 a 4 and we have 6
successively: 3 a 3 5 a 4 a 4 x + y 3z + aw 4 3x y + 5z + aw 4x + y + z + aw 3r + r r 4r + r 3 r 3 3e + e e 4e + e 3 e 3 3 a 7 4 7a 7 4 7a 4 4 4 x + y 3z + aw 4 7y + 4z + 7aw 4 7y + 4z + 7aw 4 r + r 3 r 3 e + e 3 e 3 3 a 7 4 7a 4 4 x + y 3z + aw 4 7y + 4z + 7aw 4 7 r 7 e 3 a a 4 x + y 3z + aw 4 y z aw r + r r e + e e 3a a x + z + 3aw y z aw x +z +3aw Thus, the equivalent system is and its solutions are y z aw x t 3as, y + t + as, z t, w s for s, t R Gaussian elimination Definition A matrix is said to be in reduced row-echelon form if it has the following properties: If a row does not consist entirely of zeros, then the first nonzero number in the row is a, called a leading 7
If there are any rows that consist entirely of zeros, then they are grouped together at the bottom of the matrix 3 In any two successive rows that do not consist entirely of zeros, the leading in the lower row occurs farther to the right than the leading in the higher row 4 Each column that contains a leading has zero everywhere else A matrix satisfying just the first 3 properties, namely, and 3, is said to be in row-echelon form We are now going to provide a number of steps in order to reduce a matrix to a (reduced) rowechelon form These are: Locate the lefmost column that does not consist entirely of zeros; Interchange the top row with another row, if necessary, to bring a nonzero entry to the top of the column found in Step ; 3 If the entry that is now at the top of the column found in Step is a, multiply the first row by a in order to introduce a leading ; 4 Add suitable multiples of the top row to the rows below so that all entries below the leading become zeros; 5 Cover the top row in the matrix and begin again with the Step applied to the submatrix that remains Continue in this way until the entire matrix is in row-echelon form 6 Once you got the matrix in row-echelon form, beginning with the last nonzero row and working upward, add suitable multiples of each row to the rows above to introduce zeros above the leading s The variables corresponding to the leading s in row echelon form are called leading variables and the others are called free variables The above procedure for reducing a matrix to reduced row-echelon form is called Gauss-Jordan elimination If we use only the first five steps, the procedure produces a row-echelon form and is called Gaussian elimination Examples Solve the linear system x y + z w x + y z w x + y 4z + w 3x 3w 3 () 8
Solution: The augmented matrix is and it can be reduced at 4 3 3 3 reduced row-echelon form as follows: r +r r r + r 3 r 3 3 6 3 r 4 3r + r 4 r 4 3 3 3 3 6 3 6 r + r 3 r 3 3r + r 4 r 4 r + r r such that the corresponding linear system, equivalent with the initial one, is x t x w y z and its solutions are y s z s, s, t R, w t being infinitely many in this case It is sometimes preferable to solve a linear system by using Gauss elimination procedure for the augmented matrix to bring it just in a row-echelon form and to use the so called back-substitution method afterward For the linear system, a row-echelon form of the augmented matrix is and its corresponding linear system is x y + z w y z The back-substitution method consists in the following three steps: (a) Solve the equations for the leading variables in terms of the free variables; (b) Beginning with the bottom equation and working upward, successively substitute each equation into all the equations above it 9
(c) Assign arbitrary values to the free variables, if any In our particular case x, y are the leading variables and z, w are the free variables Therefore the step (a) is The step (b) is x y z + w y z x z z + w y z or equivalently x w y z Finally, the step (c) consists in assigning to the free variables z, w the arbitrary values z s and w t such that the solutions of the initial linear system are: x t y s z s w t, s, t R Solve the linear system x + 7y z 3x y + z 4 x + 6y z 5 7 Solution: The augmented matrix is 3 4 and it can be reduced at reduced 6 5 row-echelon form as follows: 7 7 7 r 3r + r r r + r 3 r 3 3 4 3 4 5 7 r + r 3 r 3 6 5 6 5 5 7 7 5 7 /r 7 4 7 7r + r r 4 4 such that the corresponding linear system, equivalent with the initial one, is 9 7/ x + 4 z 9 y 4 z 7/ and its solutions are x 9 5t y 7+5t z t t R
3 Solve the linear system x x + x 3 4x 4 x + 3x + 7x 3 + x 4 x x x 3 6x 4 5 4 Solution: The augmented matrix is 3 7 and it can be reduced at 6 5 reduced row-echelon form as follows: 4 4 r 3 7 + r r r 5 6 6 + r 3 r 3 r + r 3 r 3 6 5 4 4 r + r 3 r 3 5 6 6 Thus the system is inconsistent, that is it has no solutions at all, since 6 6 4 For which values of a R the following linear system has no solution? Exactly one solution? Infinitely many solutions? When the system is consistent, find its solution set x y z x 4y 6z 6 3x 5y + (a 4)z a 4 Solution: The augmented matrix is 4 6 6 and it will be reduced by 3 5 a 4 a 4 performing successively the following row-operations: r 4 8 6 + r r 4 6 r 3r 3 5 a + r 3 r 3 4 a 4 a 8 a 4 r a 8 r 3 + r 3 r 3 3 a 4 a 4 a + }{{} ( ) a 4 if a {±}
a 4 if a {±} 3 a r + r r if a {±} 3 3a 8 a r 3 +r r r 3 + r r if a {±} a a 3 a a r + r r if a {±} Therefore for a {±} the given linear system has the unique solutions x 3, y 3a 8 a, z a If a, the ( ) reduced matrix becomes r 3 + r r and in this case the given linear system has infinitely many solutions 3 3, x 3, y 3 s, z s, s R Finally, when a, the ( ) reduced matrix becomes solutions at all in this case 3 4 and linear system has no 3 Matrices and matrix operations Definition 3 A matrix A a a a n a a a n a m a m a mn row row row m col col coln equally written [a ij ] m n (or simply [a ij ]), is a rectangular array of numbers The numbers in the array are called entries The entry in row i and column j of a matrix A is commonly denoted by (A) ij, in our particular case (A) ij a ij The size of a matrix is described in terms of the number of rows and the number of columns it has The above matrix A has size m n A matrix x x x x m
with only one column is called a column matrix (or a column vector), and a matrix [ y y y y n ] with only one row is called row matrix (or a row vector) A matrix a a a n a A a a n, a n a n a nn with n rows and n columns, that is of size n n, is called a square matrix of order n and the entries a, a,, a nn are said to be on the main diagonal of A Definition 3 Two matrices are said to be equal if they have the same size and their corresponding entries are equal Definition 33 If A a a a n a a a n, B b b b n b b b n, a m a m a mn b m b m b mn are matrices of the same size, then their sum is the matrix a + b a + b a n + b n a A + B + b a + b a n + b n, and their difference is the matrix A B a m + b m a m + b m a mn + b mn a b a b a n b n a b a b a n b n a m b m a m b m a mn b mn One can shortly define the entries of the sum and the difference matrices in the following way: (A + B) ij (A) ij + (B) ij and (A B) ij (A) ij (B) ij 3
If the matrices A and B have different sizes, then their sum and difference is not defined If c is any scalar (number), then the product ca is the matrix ca ca ca n ca ca ca n ca m ca m ca mn obtained by multiplying each entry of A with c and its entries can be shortly written as (ca) ij c(a) ij If A, A,, A n are matrices of the same size and c, c,, c n are scalars, then the matrix c A + c A + + c n A n is defined and it is called a linear combination of A, A,, A n with coefficients c, c,, c n Examples 34 Consider the matrices A 3, B 4 3, C 3 4 5, D 6 3 4 Compute, when is possible, A B, C + D, B C, A + C, Solution: A B 3 5 4 3 4 4 3 6 8 D + E 4 + 3 7 5 4 3 7 5 B C and A + C are not defined Compute, when possible, 3A, ( )B, A B, 3C + D, 5B + 6D Solution: 3 A 3, ( )B 4 3 3 3 A B 3 6 8 4 3 4 4 4 3 6 4 3 6 3 6 9 6 7 3C+D 3 4 + 3 3 + 3 5 5 4 6 5 3 4 7 7 7 5B + 6D is not defined 4
Definition 35 Lat A be an m n matrix and B be an n p matrix Then the product AB is the m p matrix C whose entry c ij in row i and column j is obtained as follows: Sum the products formed by multiplying, in order, each entry in row i of the matrix A with the corresponding entry in column j of the matrix B More precisely c ij a i b j + a i b j + + a in b nj n a ik b kj k A B C m n n p m p must be the same size of product AB a a a n a a a n a i a i a in a m a m a mn b b b j b p b b b j b p b n b n b nj b np n a k b k k n a k b k k n a ik b k k n a mk b k k n a k b k k n a k b k k n a k b kj k n a k b kj k n a k b kp k n a k b kp k n n n a ik b k a ik b kj a ik b kp k k k n n n a mk b k a mk b kj a mk b kp k Definition 36 The transpose of an m n matrix A, denoted by A T, is the n m matrix whose i th row is the i th column of A Thus (A T ) ij (A) ji Observe that for any matrix A we have (A T ) T A Example 37 If A 3 matrices AB, BA, (AB) T, (BA) T, A T B T, B T A T k k and B 3, find, when possible, the 4 3 5
Solution: then we have AB 3 3 }{{} 4 3 3 }{{} 3 4 + 3 + ( ) 4 + ( ) + ( ) 3 + ( ) + ( ) + + ( ) + 3 + 3 4 + ( ) + 3 3 + ( ) + 3 + + 3 8, such that (AB) T 7 8 7 4 3 4 }{{} 4 3 }{{} 4 Observe that the product BA, (BA) T, A T B T are not defined However, observe that 3 4 B T 3 and A T, 3 }{{}}{{} 3 4 3 such that the product B T A T is also defined and 3 4 8 B T A T 3 7 (AB) T 4 3 }{{} 3 }{{} 3 }{{} 4 3 4 Definition 38 If A is a square matrix, then the trace of A, denoted by tr(a), is defined to be the sum of the entries on the main diagonal, namely tr(a) : (A) + (A) + + (A) nn, where n is the order of A The trace of A is not defined if A is not a square matrix Example 39 Find, when possible, tr(aa T ) and tr(ab), where A and B 3 3 4 3 6
We first observe that AA T 3 3, such that tr(aa T ) 3 3 3 }{{} + 3 5 Since AB is not a square matrix, its trace is not defined } 8 7 {{ 4 3 } 4 If A ij, i m, j n are matrices of suitable sizes, then we can form a new matrix A A A n A A A A n A m A m A mn and call partition of the matrix A the above kind of subdivision For example - - 3 - - 3-5 A A A A A 3 A 3 5-7 6 where A, A 3, A 3 3 5, A 5, A 7, A 3 6 An application of matrix multiplication operation consists in transforming a system of linear equations into a matrix equation Indeed, for a linear system we have successively: a x + a x + + a n x n b a x + a x + + a n x n b a x + a x + + a n x n a x + a x + + a n x n b b a m x + a m x + + a mn x n b m a m x + a m x + + a mn x n b m 7
a a a n a a a n x x b b AX B The matrix } a m a m {{ a mn }} x n {{ } A X a a a n a A a a n b m } {{ } B a m a m a mn is called the coefficient matrix of the linear system Observe that the augmented matrix of the [ ] system is A B 4 Inverses Rules of matrix arithmetic Theorem 4 Assuming that the sizes of matrices are such that the indicated operations can be performed, the following rules of matrix arithmetic are valid: A + B B + A; (A + B) + C A + (B + C); 3 (AB)C A(BC); 4 A(B + C) AB + AC; 5 (A + B)C AC + BC; 6 A(B C) AB AC; 7 (A B)C AC BC; 8 a(b + C) ab + ac; 9 a(b C) ab ac; (a + b)c ac + bc; (a b)c ac bc; a(bc) (ab)c; 8
3 a(bc) (ab)c B(aC) Remark 4 The matrix multiplication is not commutative Indeed, for example if A, B, 3 3 then AB 3 3 4 3 6 3 3 3 BA Define the m n zero matrix to be the matrix O m n : denoted by O when the size is understood from context and it will be equally m n Theorem 43 Assuming that the sizes of the matrices are such that the indicated operations can be performed, the following rules of matrix arithmetic are valid A + O O + A A; A A O; 3 O A A; 4 AO O The identity matrix of order n is defined to be the square the n n matrix I n :, and it will be equally denoted by I when the order is understood from context Remark 44 If A is an m n matrix, then AI n I m A A n n Theorem 45 If R is the reduced row-echelon form of an n n matrix A, then either R has a row of zeros, or R is the identity matrix 9
Definition 46 A square matrix A is said to be invertible if there is another square matrix B, of the same size, such that AB BA I Remark 47 If B, C are both inverses of a A, then B C Indeed, we have successively: B BI B(AC) (BA)C IC C Therefore, the inverse of an invertible matrix A is unique and denoted by A, being characterized by the equalities: AA A A I Example 48 If a, b, c, d R are such that ad bc, then the matrix A invertible and A ad bc a b } c d {{ } A d b c a d ad bc b ad bc c ad bc a ad bc d ad bc b ad bc c ad bc ad bc ad bc cd dc ad bc a ad bc Indeed we have ab+ba ad bc cb+da ad bc a c b d is and similarly d ad bc b ad bc c ad bc a ad bc a c b d } {{ } A da bc ad bc ca+ac ad bc db bd ad bc cb+ad ad bc If A is a square matrix, then its positive powers are defined to be A I, A n AA }{{ A} for n > n times Moreover, if A is invertible, then A n A} A {{ A }, also for n > n times Theorem 49 If A is an invertible matrix, then A is also invertible and (A ) A; A n is invertible and (A n ) (A ) n for n {,,, }; 3 A T is invertible and (A T ) (A ) T ; 4 For any non-zero scalar k, the matrix ka is invertible and (ka) k A ; 5 A m A n A m+n for any integers m, n; 6 (A m ) n A mn for any integers m, n Theorem 4 If the sizes of the involved matrices are such that the stated operations can be performed, then
(A T ) T A; (A + B) T A T + B T and (A B) T A T B T ; 3 (ka) T ka T, where k is any scalar; 4 (AB) T B T A T Example 4 Find the matrix A knowing that (I + A) 4 5 Solution: Since (I + A), it follows that [(I + A) ] 4 5 4 5 Therefore we have successively I + A 5 ( ) 5 4 4 I + A 3 5 4 A 3 5 4 A 5 3 3 4 3 3 A 8 3 3 4 3 3 A 8 3 3 4 3 3 A 8 3 3 4 3 3 A 9 3 3 3 6 3 If A cos θ sin θ sin θ cos θ, show that A cos θ sin θ sin θ cos θ and A 3 cos 3θ sin 3θ sin 3θ cos 3θ Solution: Indeed, A cos θ sin θ sin θ cos θ cos θ sin θ sin θ cos θ cos θ sin θ sin θ cos θ sin θ cos θ cos θ sin θ cos θ sin θ sin θ cos θ
A 3 AA cos θ sin θ sin θ cos θ cos θ sin θ sin θ cos θ cos θ cos θ sin θ sin θ (cos θ sin θ + sin θ cos θ) sin θ cos θ + cos θ sin θ sin θ sin θ + cos θ cos θ cos(θ + θ) sin(θ + θ) sin(θ + θ) cos(θ + θ) cos 3θ sin 3θ sin 3θ cos 3θ 5 Elementary matrices and a method for finding A Definition 5 An n n matrix is called an elementary matrix if it can be obtain from the n n identity matrix I n by performing a single elementary row operation Example 5 3r I 3 + r 3 r 3 E elementary matrix 3 3 3 3 If A 6 3, then EA 6 3 r 3 r 3 +r 6 3 4 8 9 4 A Theorem 53 If the elementary matrix E results from performing a certain row operation on I m and A is an m n matrix, then the product EA is the matrix that results when the same row operation is performed on A If an elementary operation is performed on the identity matrix I to obtain an elementary matrix E, then there is another operation, called the corresponding inverse operation, when apply to E to obtain I back again Direct Operation Corresponding Inverse operation Multiply row i by c (cr i ) Multiply row i by c ( c r ) i Interchange rows i and j (i j) Interchange rows i and j (i j) Add c times row i to row j (cr i + r j r j ) Add c times row i to row j ( cr i + r j r j )
Theorem 54 Every elementary matrix is invertible, and the inverse is also an elementary matrix Proof If I dir op E inv op I Dacă I inv op E, then E E I and EE I, namely E is invertible Theorem 55 If A is an n n matrix, then the following statements are equivalent: A is invertible; The homogeneous linear system AX O has only the trivial solution; 3 The reduced row-echelon form of A is I n ; 4 A is expressible as a product of elementary matrices The last statement, (4), of theorem 67 is of particular importance since it provide us a method of finding the reduced row-echelon form of a matrix and a method of finding the inverse of an invertible matrix Let A be a matrix and R be its reduced row-echelon form, namely A op op A op 3 op A n R, then R E n E A, where I op E, I op E,, I op n E n Therefore A (E n E ) R E E n R If A is particularly invertible, then R I, such that A E E and implicitly n A E n E Thus, in order to find the inverse of an invertible matrix A, we first observe that we have the general property A[BC] [ABAC] and that we have successively: [AI] op E [AI] [E AE I] op E [E AE I] [E E AE E I] op 3 op n [E n E A }{{} I E n E I] }{{} A 4 Examples 56 Find the inverse of the matrix A 3 7 and write A as a 5 product of elementary matrices; 3
Express the matrix 7 8 A 3 3 8 5 8 in the form A EF GR, where E, F, G are elementary matrices and R in row-echelon form () 4 3 r 3 7 + r r 3r 3 7 + r r r + r 3 r 3 5 5 3 5 5 3 r 3 3 + r r 3r 3 + r r }{{} A Therefore E 5 E 4 E 3 E E A I 3, where r I 3 + r r : E 3r I 3 + r r 3 : E r I 3 + r 3 r 3 : E 3 r I 3 3 + r r : E 4 3 3r I 3 3 + r r : E 5 () 7 8 3 3 8 r A 3 3 8 r r 7 8 + r 3 r 3 5 8 5 8 4
r + r 3 r 3 3 3 8 7 8 7 8 r + r 3 r 3 3 3 8 7 8 R matrix in row-echelon form Consequently R E 3 E E A, or equivalently A E E E R, where 3 r I 3 r : E E I 3 r + r 3 r 3 r + r 3 r 3 :E :E I 3 r + r 3 r 3 r + r 3 r 3 :E 3 :E 3 Now it is enough to take E E, F E, G E 3 6 Further results on systems of equations and invertibility Theorem 6 Every system of linear equations has either no solution, exactly one solution or infinitely many solutions Theorem 6 If A is an invertible n n matrix, then for each n matrix b, the system of linear equations AX B has exactly one solution, namely X A B Example 63 Find the solution set of the following linear system: x y + 4z 3x + y 7z x 5z 5
Solution: We first observe that the coefficient matrix of the given linear system is A 4 5 5 3 3 7, which is invertible and its inverse is A The given linear 5 x system can be written in the form AX B, where X y and B Consequently the z 5 5 3 unique solution of the given linear system is x A B Frequently, one have to solve a sequence of linear systems AX B, AX B,, AX B k each of which has the same square matrix A If A is invertible then the solutions are X A B, X A B,, X k A B k Otherwise the method of solving those systems, which works both for A invertible or A noninvertible, consists in forming the matrix [A B B B k ] and by reducing it to the reduced row-echelon form Example 64 Solve the following linear systems: x y + 4z 3x + y 7z x 5z, x y + 4z 3x + y 7z x 5z 3 Theorem 65 Let A be a square matrix If B is a square matrix satisfying BA I, then B A ; If B is a square matrix satisfying AB I, then B A Theorem 66 Theorem 67 If A is an n n matrix, then the following statements are equivalent: A is invertible (*); 6
The homogeneous linear system AX O has only the trivial solution; 3 The reduced row-echelon form of A is I n ; 4 A is expressible as a product of elementary matrices 5 AX B is consistent for every n matrix B;(*) 6 AX B has exactly one solution for every n matrix B(*) Theorem 68 Let A and B be square matrices of the same size If AB is invertible, then A and B are both invertible Fundamental Problem 69 Let A be a fixed m n matrix Find all m matrices B such that the system of equations AX B is consistent Example 6 What conditions must a, b, b, b 3 satisfy in order for the system of the linear system x + y z b x y + z b to be consistent? x + y + z b 3 Solution: For the augmented matrix of the given linear system we have successively: b a b r + r r r + r r b 3 3 b r + r 3 r 3 b b 3 3 8 b 3 + b a b r + r r 3 3 b b b 3 + b + b Thus, the given linear system is consistent if and only if b 3 + b + b 7 Diagonal, Triangular and Symmetric Matrices Definition 7 A square matrix in which all the entries off the diagonal are zero is called a diagonal matrix The general form a of a diagonal matrix is d d D d n 7
Remarks 7 The diagonal matrix D d d d n is invertible iff all of its diagonal entries are non-zero In this case d D d d n Powers of diagonal matrices are easy to be computed More precisely, if d d k d D then D k d k d n d k n 3 If A [a ij ] is an m n matrix, then δ a a a n δ a a a n δ a δ a δ a n δ a δ a δ a n δ m a m a m a mn δ m a m δ m a m δ m a mn and a a a n a a a n d d a d a d a n d n a d a d a n d n a m a m a mn d n a m d a m d a mn d n Examples 73 Find a diagonal matrix A satisfying 9 A 4 8
Solution: We have successively: A 9 4 [A ] 9 4 A 9 4 A ( 3 ) ( Thus, such a matrix is A 3 Definition 74 A square matrix in which all the entries above the main diagonal are zero is called lower triangular and a square matrix in which all the entries below the main diagonal are zero is called upper triangular A matrix which is either lower triangular or upper triangular is called triangular The general form a of a lower triangular matrix is a a a a n a n a nn and the general form a of a upper triangular matrix is a a a n a a n a nn Remark 75 A square matrix A [a ij ] is upper triangular iff a ij for i > j; A square matrix A [a ij ] is lower triangular iff a ij for i < j Theorem 76 The transpose of a lower triangular matrix is upper triangular, and the transpose of an upper triangular matrix is lower triangular; The product of lower triangular matrices is a lower triangular matrix; 9
3 The product of upper triangular matrices is a upper triangular matrix; 4 A triangular matrix is invertible if and only if its diagonal entries are all nonzero; 5 The inverse of a lower triangular matrix is lower triangular and the inverse of an upper triangular matrix is upper triangular Example 77 If A 3 and B 3 5, show that AB is also upper triangular and find (AB) and B A Solution: 3 4 9 3 9 5 5 9 4 9 9 R, R, 5 R3 9 5 5 9 9 9 R + R 3 R 5 Thus 9 (AB) B A 9 5 R + R R 9 R + R R 3 9 9 5 Definition 78 A square matrix A is called symmetric if A T A, or equivalently (A) ij (A) ji Theorem 79 If A, B are symmetric matrices of the same size and k is any scalar, then: A T is symmetric; A + B and A B are symmetric; 3 ka is symmetric Theorem 7 matrices commute; The product of two symmetric matrices is symmetric if and only if the 3
If A is an invertible symmetric matrix, then A is symmetric Theorem 7 If A is an arbitrary matrix, then AA T and A T A are symmetric matrices; If A is an invertible matrix, then AA T and A T A are invertible matrices Example 7 Find the values of a, b, c R for which the matrix A is symmetric, where a + b + c 4b 3c A 5 a + c 4 6 Solution: The required values are solutions of the linear system a + b + c 5 4b 3c a + c 4 can be solved by performing row-reduction operations in the corresponding augumented matrix, namely: 5 5 r +r 3 r 3 4 3 4 4 3 r 4 r 4 6 7 9 5 5 3 r +r 3 r 3 4 3 5 4 r 4 3 3 4 The corresponding linear system of the last matrix in the above row-reduction process, which is 6 5 5 equivalent with the initial one, has the unique solution a, b c that 3
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Chapter Determinants The determinant function Definition A permutation of the set {,,, n} is a bijective function σ : {,,, n} {,,, n}, namely an arrangement of the integers,,, n without omissions or repetitions It is also written σ n σ() σ() σ(n) An inversion of the permutation σ is a pair (i, j) such that i < j and σ(i) > σ(j) Denote by m(σ) the number of inversions of σ and by ε(σ) the signature ( ) ε(σ) of σ A permutation σ is called even/odd if m(σ) is even/odd, or equivalently ε(σ) +/ε(σ) Definition The determinant of a square n n matrix A [a ij ] is defined to be det(a) ε(σ)a σ() a σ() a nσ(n) It is denoted either by det(a) or by a a a n a a a n a n a n a nn Remarks 3 If A a a is a square matrix, then det(a) a a a a a a Indeed, the only permutations are e and σ and ε(e), 33
ε(σ) since m(e) and m(σ) Thus det a a a a a a a a ε(e)a a + ε(σ)a a a a a a e() e() σ() σ() For example 3 ( )3 3 + 3 6 3 A matrix A a a a a is invertible if and only if det(a) In this case A a a det(a) a a a a a 3 3 If A a a a 3 is a square matrix, then a 3 a 3 a 33 a a a 3 det(a) a a a 3 a 3 a 3 a 33 a a a 33 +a a 3 a 3 +a 3 a a 3 a 3 a a 3 a a a 33 a a 3 a 3 a a a a a 3 a a a a a 3 a 3 a 3 a 33 a 3 a 3 + + + Indeed, the only 3 3 square matrices are e 3, σ : 3, σ 3, 3 3 3 σ 3 3, σ 4 3, σ 5 3 3 3 3 and ε(e) ε(σ 4 ) ε(σ 5 ) while ε(σ ) ε(σ ) ε(σ 4 ) since m(e), m(σ 4 ) m(σ 5 ) and m(σ ), m(σ ) 3, m(σ 3 ) For example 3 3 6 + + ( 8) 8 + 8 6 34
Evaluating determinants by row reduction Theorem Let A be a square matrix If A has a row of zeros, then det(a) det(a) det(a T ) Theorem If A is an n n triangular matrix, then det(a) is the product of the entries on the main diagonal, namely det(a) a a a nn Theorem 3 Let A be an n n square matrix If B is the matrix that results when a single row or a single column of A is multiplied by a scalar k, then det(b) k det(a); Consequently det(ka) k n det(a) If B is the matrix that results when two rows or two columns of A are interchanged, then det(b) det(a); 3 If B is the matrix that results when a multiple of one row of A is added to another row or when a multiple of one column of A is added to another column, then det(b) det(a) Corollary 4 If A is a square matrix with two proportional rows or two proportional columns, then (A) Example 5 Find the determinant of A 3 4 Solution: 3 4 4 ( ) det(a) 3 4 r + r r 3 4 ( ) ( + ) 4 35
3 Properties of determinant function Theorem 3 Let A, B and C be n n matrices that differ only in a single row, say the r th, and assume that the r th row of C can be obtained by adding corresponding entries in the r th row of A and B Then det(c) det(a) + det(b) In other words a a a n a a a n a + a a + a a n + a n a m a m a mn a a a n a a a n a a a n a m a m a mn + a a a n a a a n a a a n a m a m a mn Theorem 3 If A and B are square matrices of the same size, then det(ab) det(a) det(b) Theorem 33 If A is an invertible matrix, then det(a ) det(a) Theorem 34 If A is an n n matrix, then the following statements are equivalent: A is invertible;(*) The homogeneous linear system AX O has only the trivial solution; 3 The reduced row-echelon form of A is I n ; 4 A is expressible as a product of elementary matrices 5 AX B is consistent for every n matrix B; 6 AX B has exactly one solution for every n matrix B 7 det(a) (*) Example 35 Consider the matrix A for all x, y, z, w R x y z w w x y z y + z z + w w + x x + y Show that det(a) Solution: For the determinant of A we have successively: x y z w r +r 4 r 4 x y z w w x y z w x y z y + z z + w w + x x + y x + y + z y + z + w z + w + x w + x + y 36
r 3 +r 4 r 4 x y z w w x y z x + y + z + w x + y + z + w y + z + w + x z + w + x + y (x + y + z + w) x y z w w x y z for all x, y, z, w R Let A a b c d e f g h i Assuming that det(a) 7, find det(3a), det(a ), det(a ) and det a g d b h e c i f Solution: det(3a) 3 3 det(a) 7( 7) det(a ) det(a) 7 7 ( ) det(a ) 3 det(a ) 8 7 8 7 det a g d b h e c i f det a g d b h e c i f T a b c g h i d e f a b c d e f g h i ( 7) 7 3 Show that a + b t a + b t a 3 + b 3 t a t + b a t + b a 3 t + b 3 c c c 3 ( t ) a a a 3 b b b 3 c c c 3 37
Solution: Indeed, we have successively: a + b t a + b t a 3 + b 3 t a t + b a t + b a 3 t + b 3 c c c 3 a a a 3 a t + b a t + b a 3 t + b 3 c c c 3 + b t b t b 3 t a t + b a t + b a 3 t + b 3 c c c 3 a a a 3 a t a t a 3 t c c c 3 }{{} + a a a 3 b b b 3 c c c 3 + b t b t b 3 t a t a t a 3 t c c c 3 + b t b t b 3 t b b b 3 c c c 3 }{{} a a a 3 b b b 3 c c c 3 + t b b b 3 a a a 3 c c c 3 a a a 3 b b b 3 c c c 3 t a a a 3 b b b 3 c c c 3 ( t ) a a a 3 b b b 3 c c c 3 4 Cofactor expansion Cramer s rule Definition 4 If A is a square matrix, then the minor of entry a ij is denoted by M ij and is defined to be the determinant of the submatrix that remains after the i th row and the j th column are deleted from A The number ( ) i+j M ij is denoted by C ij and is called the cofactor of entry a ij Example 4 a a a 3 a a a 3 a 3 a 3 a 33 a a a 33 + a a 3 a 3 + a 3 a a 3 a 3 a a 3 a a a 33 a a 3 a 3 a (a a 33 a 3 a 3 ) a (a a a 33 a 3 a 3 ) + a 3 (a a 3 a a 3 ) ( ) + a a a 3 a 3 a 33 + ( )+ a a a 3 a 3 a 33 + ( )+3 a 3 a a a 3 a 3 a C + a C + a 3 C 3 38
Similarly a a a 3 a a a 3 a 3 a 3 a 33 a a a 33 + a a 3 a 3 + a 3 a a 3 a 3 a a 3 a a a 33 a a 3 a 3 a (a a 33 a 3 a 3 ) + a (a a 33 a 3 a 3 ) a 3 (a a 3 a 3 a ) ( ) + a a 3 a 3 a 33 + ( )+ a a 3 a 3 a 33 + ( )3+ a a 3 a a 3 a C + a C + a 3 C 3 Theorem 43 If A [a ij ] is a square n n matrix, then det(a) a i C i + a i C i + + a in C in }{{} The cofactor expansion of det(a) along the i th row a j C j + a j C j + + a nj C nj }{{} The cofactor expansion of det(a) along the j th column Example 44 Show that: a a a 3 a 4 a a a 3 a 4 a 3 a 3 a 3 3 a 3 4 (a a )(a a 3 )(a a 4 )(a a 3 )(a a 4 )(a 3 a 4 ) If A [a ij ] is a square n n matrix and C ij is the cofactor of the entry a ij, then the matrix C C C n C C C n C n C n C nn is called the matrix of cofactors from A The transpose of this matrix is called the adjoint of A and is denoted by adj(a) If A is an invertible matrix, then A det(a) adj(a) The inverse of an invertible lower triangular matrix is lower triangular and the inverse of an invertible upper triangular matrix is upper triangular Find necessary and sufficient conditions on a, b, c R A a a a a a a such that the matrix A is invertible In those cases find A 39
According to theorem 34, A is invertible if and only if det(a) But since det(a) V (a, b, c) a b c (c a)(c b)(b a), a b c it follows that A is invertible if and only if a b, b c and c a In this case C A det(a) adj(a) C C 3 C det(a) C C 3 T C 3 C 3 C 33 (c a)(c b)(b a) b c b c b c b c a c a c a c a c a b a b a b a b T (c a)(c b)(b a) bc(c b) ac(c a) ab(b a) (c b ) c a (b a ) c b (c a) b a T (c a)(c b)(b a) bc(c b) b c c b ac(a c) c a a c ab(b a) a b b a bc(c b) (c a)(c b)(b a) ac(a c) (c a)(c b)(b a) ab(b a) (c a)(c b)(b a) (b c)(b+c) (c a)(c b)(b a) (c a)(c+a) (c a)(c b)(b a) (a b)(a+b) (c a)(c b)(b a) c b (c a)(c b)(b a) a c (c a)(c b)(b a) b a (c a)(c b)(b a) bc (c a)(b a) ac (c b)(a b) ab (c a)(c b) b+c (a c)(b a) c+a (c b)(b a) a+b (c a)(b c) (c a)(b a) (c b)(a b) (c a)(c b) (Cramer s Rule) If AX B is a system of n linear equations in n unknowns such that det(a), then the system has a unique solution This solution s where A j matrix B x det(a ) det(a), x det(a ) det(a),, x n det(a n) det(a), is the matrix obtained from A by replacing its j th row with the entries of the column Provide necessary and sufficient conditions on a, a, a 3, a 4 R such that the linear 4
system x + y + z + w ax + by + cz + dw α a x + b y + c z + d w α a 3 x + b 3 y + c 3 z + d 3 w α 3 () has exactly one solution for each α R In this case solve the system by using the Cramer s rule The given linear system has a unique solution for each α R if and only if its coefficient matrix A is invertible, or, equivalently det(a) But since a b c d a b c d a 3 b 3 c 3 d 3 det(a) V (a, b, c, d) (d a)(d b)(d c)(c a)(c b)(b a), it follows that the given linear system has unique solution for each α R if and only if the scalars a, b, c, d are pairwise disjoint By Crammer s rule, it follows that the unique solution of the system is x y z w V (α, b, c, d) V (a, b, c, d) V (a, α, c, d) V (a, b, c, d) V (a, b, α, d) V (a, b, c, d) V (a, b, c, α) V (a, b, c, d) (d α)(c α)(b α) (d a)(c a)(b a) ; (α a)(c α)(d α) (b a)(c b)(d b) ; (α a)(α b)(d α) (c a)(c b)(d c) ; (α a)(α b)(α c) (d a)(d b)(d c) 4
4
Chapter 3 Euclidean vector spaces 3 Euclidean n-space Definition 3 If n is a positive integer, then an ordered n-tuple is a sequence of n numbers (a, a,, a n ) The set of R n of all n-tuples is called n-space A -tuple is also called ordered pair and a 3-tuple is called ordered triple and both of them have geometric interpretation Thus an n-tuple can be viewed as either an generalized point or a generalized vector Definition 3 Two vectors u (u, u,, u n ) and v (v, v,, v n ) in R n are called equal if u v, u v,, u n v n Their sum u + v is defined by u + v (u + v, u + v,, u n + v n ) and, for a sscalar k, the scalar multiple is defined by ku (ku, ku,, ku n ) The zerovector in R n is denoted by and is defined to be the vector (,,, ) If u (u, u,, u n ) R n is any vector, the negative or additive inverse of u is denoted by u and is defined by u ( u, u,, u n ) The difference u v of the vectors (u, u,, u n ), v (v, v,, v n ) R n is defined by u v : u + ( v) In termes of components we have u v (u v, u v,, u n v n ) Theorem 33 If u (u, u,, u n ), v (v, v,, v n ), w (w, w,, w n ) R n and k, l are scalars, then u + v v + u u + (v + w) (u + v) + w 3 u + + u u 4 u + ( u), namely u u 43
5 k(lu) (kl)u 6 k(u + v) ku + kv 7 (k + l)u ku + lu 8 u u Definition 34 If u (u, u,, u n ), v (v, v,, v n ) R n are any vectors, then the Euclidean Product u v is defined by u v : u v + u v + + u n v n Theorem 35 If u (u, u,, u n ), v (v, v,, v n ), w (w, w,, w n ) R n and k, l are scalars, then u v v u k(u v) (ku) v u (kv) 3 (u + v) w u w + v w 4 u u and u u if and only if u Definition 36 The Euclidean norm or the Euclidean length u of a vector u (u, u,, u n ) R n is defined by u : u u u + u + + u and the distance d(u, v) n between the points u (u, u,, u n ), v (v, v,, v n ) R n is defined by d(u, v) u v (u v ) + (u v ) + + (u n v n ) Theorem 37 (Cauchy-Schwarz Inequality in R n ) If u (u, u,, u n ), v (v, v,, v n ) R n, then u v u v In terms of components, the inequality becomes u v + u v + + u n v n u + u + + u v + n v + + v n Theorem 38 If u, v R n and k is any scalar, then u u if and only if u 3 ku k u 44
4 u + v u + v (Triangle inequality) Theorem 39 If u, v, w R n and k is any scalar, then d(u, v) d(u, v) if and only if u v 3 d(u, v) d(v, u) 4 d(u, v) d(u, w) + d(w, v) (Triangle inequality) Theorem 3 If u, v R n, then u v 4 u + v 4 u v Proof Indeed, by adding the identities u + v (u + v) (u + v) u + u v + v u v (u v) (u v) u u v + v, we immediately get the required identity Definition 3 Two vectors u, v R n are said to be orthogonal if u v Theorem 3 (Theorem of Pythagoras for R n ) If u, v R n are orthogonal vectors, then the equality u + v u + v holds Proof Indeed, we have successively: u + v (u + v) (u + v) u + u v + v u + v A vector u (u,, u n ) R n can be naturally identified with a column or with a row matrix, namely with u u u or u [u u u n ] u n Consequently, the set R n of n-tuples can be naturally identified with either the space of column matrices, or with the space of row matrices The sum u + v of two vectors u (u,, u n ), v (v,, v n ) will be identified consquently identified with the sum u v u + v u u + v v + u + v u n v n u n + v n 45
of the column matrices u and v, or with the sum u + v [u u u n ] + [v v v n ] [u + v u + v u n + v n ] of the row matrices u and v Similarly, the scalar multiple ku of the real k with the ordered n-tuple u (u, u,, u n ) can be either identified with the scalar multiple ku k u u u n ku ku ku n, or with the scalar multiple ku k[u, u,, u n ] [ku, ku, ku n ] 3 Linear transformations from R n to R m Definition 3 A mapping T A : R n R m, T A (x) Ax, where A is an m n matrix, is called linear transformation, or linear operator, if m n The linear transformation T A is equally called the multiplication by A If A [a ij ] mn, then T ( x x x n ) a a a n a a a n a m a m a mn x x x n a x + a x + + a n x n a x + a x + + a n x n a m x + a m x + + a mn x n Observe that T ( ) a a a m, T ( ) a a a m,, T ( ) a n a n a mn The matrix A is called the standard matrix of the linear transformation T and it is usually denoted by [T ] and the multiplication by A mapping is also denoted by T A Therefore [T ] 46
[T (e )T (e ) T (e n )], where e, e,, e n The orthogonal projection of R on the x-axis, p x : R R, p x (x, x ) (x, ) is a linear mapping since and [p x ] ( p x x x Its equations are : ) x w x w x x The orthogonal projection of R on the y-axis, p y : R R, p y (y, y ) (, y ) is a linear mapping since and [p y ] ( p y y y Its equations are : ) y w w y y y 3 The reflection of R 3 about the x x -plane, S x x : R 3 R 3, S x x (x, y, z) (x, y, z) is a linear mapping since ( S x x x y z ) x y z x y z and [S x x ] Its equations are : w x w y w 3 z 4 The reflection of R 3 about the x x 3 -plane, S x x 3 : R 3 R 3, S x x 3 (x, y, z) (x, y, z) is a linear mapping since ( S x x 3 x y z ) x y z x y z 47
and [S x x 3 ] Its equations are : w x w y w 3 z 5 The reflection of R 3 about the x x 3 -plane, S x x 3 : R 3 R 3, S x x 3 (x, y, z) (x, y, z) is a linear mapping since x x ( ) S x x 3 y y z z w x and [S x x 3 ] Its equations are : w y w 3 z x y z 6 The orthogonal projection of R 3 on the x x -plane, P x x : R 3 R 3, P x x (x, y, z) (x, y, ) is a linear mapping since x x ( ) P x x y y z w x and [P x x ] Its equations are : w y w 3 x y z 7 The orthogonal projection of R 3 on the x x 3 -plane, P x x 3 : R 3 R 3, P x x 3 (x, y, z) (x,, z) is a linear mapping since x x ( ) P x x 3 y z z w x and [P x x 3 ] Its equations are : w w 3 z x y z 8 The orthogonal projection of R 3 on the x x 3 -plane, P x x 3 : R 3 R 3, P x x 3 (x, y, z) (, y, z) is a linear mapping since ( P x x 3 x y z ) y z x y z 48
and [P x x 3 ] Its equations are: w w y w 3 z 9 The rotation operator of R through a fixed angle θ, R θ : R R, R θ (x cos θ y sin θ, x sin θ + y cos θ) is a linear mapping since ( R x ) x cos θ y sin θ θ cos θ sin θ x y x sin θ + y cos θ sin θ cos θ y and [R θ ] cos θ sin θ w x cos θ y sin θ Its equations are: sin θ cos θ w x sin θ + y cos θ Indeed, the rotation operator R θ rotates the point (x, y) (r cos ϕ, r sin ϕ), counterclockwise with the angle θ if θ > and clockwise if θ <, the coordinates of the rotated point being w x cos θ y sin θ (w, w ) (r cos(θ + ϕ), r sin(θ + ϕ)), such that one gets w x sin θ + y cos θ The rotation operator of R 3 through a fixed angle θ about an oriented axis, rotates about the axis of rotation each point of R 3 in such a way that its associated vector sweeps out some portion of the cone determine by the vector itself an by a vector which gives the direction and the orientation of the considered oriented axis The angle of the rotattion is measured at the base of the cone and it is measured clockwise or counterclockwise in relation with a viewpoint along the axis looking toward the origin As in R, the positives angles generates counterclockwise roattions and negative angles generates clockwise roattions The counterclockwise sense of rotaion can be determined by the right-hand rule: If the thumb of the right hand points the direction of the direction of the oriented axis, then the cupped fingers points in a counterclockwise direction The rotation operators in R 3 are linear For example (a) The counterclockwise rotation about the positive x-axis through an angle θ has the w x equations w y cos θ z sin θ, its standard matrix is cos θ sin θ w 3 y sin θ + z cos θ sin θ cos θ (b) The counterclockwise rotation about the positive y-axis through an angle θ has the w x cos θ + z sin θ cos θ sin θ equations w y, its standard matrix is w 3 x sin θ + z cos θ sin θ cos θ 49
(c) The counterclockwise rotation about the positive z-axis through an angle θ has the w x cos θ y sin θ cos θ sin θ equations w x sin θ + y cos θ, its standard matrix is sin θ cos θ w 3 z (d) The homotopy of ratio k R is the linear operator H k : R n R n, H k (x) kx, Its standard matrix is k w kx k w kx and its equations are k w n kx n If k, then H k is called contraction, and for k, H k is called dilatation Let A, B be matrices of sizes n k and k m respectively, and T A : R n R k, T B : R k R m are the linear transformations of standard matrices A and B respectively, namely T A (x) Ax, T B (y) By Then the composed mapping T B T A is also linear and T B T A T BA Indeed (T B T A )(x) T B (T A )(x)) T B (Ax) B(Ax) (BA)x T BA (x) We have just proved that [T T ] [T ][T ] for any two linear mappings whose composition T T is defined Similarly, if the composed map T 3 T T, of the linear mappings T 3,T,T, is defined, then [T 3 T T ] [T 3 ][T ][T ] Let T : R R be the reflection operator about the line y x and let T : R R be the orthogonal projection about the y-axis Find T T, T T and their standard matrices Show that R θ R θ R θ R θ and it is the rotation of angle θ + θ 33 Properties of linear transformations from R n to R m Definition 33 A linear transformation T : R n R m is said to be one-to-one if u, u R n, u u T u T u It follows that each vector w in the range {T x : x R n } of an one-to-one linear transformation T : R n R m there is exactly one vector x such that T x w Theorem 33 If A is an n n matrix and T A following statements are equivalent: : R n R n is the multiplication by A, then the 5
A is invertible The range of T A is R n 3 T A is one-to-one If T A : R n R n is a one-to-one linear operator, then the matrix A is invertible and T A is also linear Moreover, : Rn R n T A (T A (x)) AA x x T I x id R n (x), x R n T A (T A (x)) A Ax x T I x id R n (x), x R n Consequently T A is also invertible and its inverse is T A T A Therefore, for the standard matrix of the inverse of an one-to-one linear operator T : R n R n, we have [T ] [T ] Examples 333 The rotation operator of R through a fixed angle θ, R θ : R R, R θ (x cos θ y sin θ, x sin θ + y cos θ) is a linear mapping since has the standard matric and [R θ ] cos θ sin θ, which is invertible sin θ cos θ and [R θ ] cos θ sin θ cos( θ) sin( θ) [R θ ] Consequently R θ is invertible sin θ cos θ sin( θ) cos( θ) and R R θ θ Theorem 334 A transformation T : R n R m is linear if and only if T (u + v) T (u) + T (v) for all u, v R n T (cu) ct (u) for all u R n Proof Indeed, if T T A, then T T A, namely T A (u + v) A(u + v) Au + Av T A u + T A v Conversely, if the given conditions are satisfied, then one can easily show that for any k vectors u,, u k R n we have T(u + + u k ) T(u ) + + T(u k ) Oncan now show that T T A, where A [T(e ) T(e n )], where e, e,, e n 5
Indeed, for x x x x n x + x + + x n x e + x e + + x n e n we have T(x) T(x e + x e + + x n e n ) x T(e ) + x T(e ) + + x n T(e n ) Ax 5