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Transcript
Chapter 10 Matrices and Linear Equations 10.1 Introduction Just as functions act upon numbers, we shall see that matrices act upon vectors and are mappings from one vector space to another. 10.2 matrices and matrix algebra A matrix is a rectangular array of quantities that are called the elements of the matrix. Let us consider the elements to be real numbers. Matrix A may be expressed as a11 a12 a a 21 22 am1 am 2 a1n a2 n amn (1) 1 A horizontal line of elements is called a row, and a vertical line is called a column. The first subscript on aij the row index, and the second subscript the column index . The matrix can be expressed in different forms. a c b d a11 a21 a12 a22 (2) aij (3) aij in Eq. (3) is called ij element and i=1,…,m and j=1,…,n. Two matrices are said to be equal if they are of the same form and if their corresponding elements are equal. Matrix addition. If aij and bij are any two matrices of the same form, say m × n, then their sum A + B is (4) aij bij and is itself an m × n matrix. 2 Scalar multiplication. If aij is any m × n matrix and c is any scalar, their product is defined as c caij , (7) THEOREM 10.2.1 Properties of Matrix Addition and Scalar Multiplication If A, B, and C are m × n matrices, O is an m × n zero matrix, and α, β , (commutativity) (9a) (A + B) + C = A + (B + C), A + 0 = A, A + (-A) = 0, ( ) ( ) , ( ) , ( ) , (associativity) (associativity) (9b) (9c) (9d) (9e) (distributivity) (distributivity) (9f) (9g) 1A = A, (9h) 0A = 0, (9i) 0 0, (9j) 3 The definitions of addition and scalar multiplication above are identical to those introduced in Sec. 9.4 for n-tuple vectors. We may refer to the matrices a11 (10) a11 , , a1n and = an1 as n-dimensional row and column vectors, respectively. Cayley product, if aij is any m × n matrix and bij is any n × p matrix, then the product AB is defined as (11) n aik bkj k 1 n cij aik bkj . If we denote AB = C = {cij} (12) k 1 If the number of columns of A is equal to the number of rows of B, then A and B are said to be conformable for multiplication; if not, the product AB is not defined. 4 It is extremely important to see that matrix multiplication is not, in general, commutative; that is, , (15) except in exceptional cases. The system of m linear algebraic equations a11 x1 a12 x2 a1n xn c1 , a21 x1 a22 x2 a2 n xn c2 , am1 x1 am 2 x2 amn xn cm (17) in the n unknowns x1, …, xn is equivalent to the single compact matrix equation Ax = c, (18) 5 a11 a12 a a 21 22 am1 am 2 a1n x1 c1 a2 n x2 c2 , x , and c . amn xn cm (19) A is called the coefficient matrix. Any n × n matrix aij is said to be square, and of order n, elements a11, a22, …, ann are said to lie on the main diagonal of A. If A is square and p is any positive integer, we defined (21) p p factors The familiar laws of exponents, p q p q , ( p )q pq follow for any positive integers p and q. (22) 6 If, in particular, the only nonzero elements of a square matrix lie on the main diagonal, A is said to be a diagonal matrix. For example, d11 0 0 d 22 D 0 0 , d nn (23) If futhermore,d11=d22=…=dnn=1, then D is called the identity matrix I. Thus, Where ij 1 0 0 1 I 0 0 ij , 1 (25) is the Kronecker delta symbol defined 1 if i j , ij 0 if i j. (26) 7 The key property of the identity matrix is that if A is any square matrix IA = AI = A, (27) A is any n × n matrix, we define 0 (28) Where I is an n × n identity matrix. THEOREM 10.2.2 “Exceptional”Properties of Matrix Multiplication (i) AB ≠ BA in general. (ii) Even if A ≠ 0, AB = AC does not imply that B = C. (iii) AB=0 does not imply that A = 0 and/or B = 0 (iv) A2 = I does not imply that A = +I or –I. 8 THEOREM 10.2.3 “Ordinary” Properties of Matrix Multiplication If α, β are scalars, and the matrices A, B, C are suitably conformable, then (αA)B = A(αB) = α(AB) (associativity) (30a) A(BC) = (AB)C, (associativity) (30b) (A+B)C = AC + BC, (distributivity) (30c) C(A + B) = CA + CB, (distributivity) (30d) A(αB + βC) = αAB + βAC (linearity) (30e) 9 Partitioning. The idea for partition is that any matrix A (which is larger that 1 × 1) may be partitioned into a number of smaller matrices called blocks by vertical lines that extend from bottom to top, and horizontal lines that extend from left to right. 2 0 3 2 0 3 5 2 7 5 2 7 11 12 21 22 , 1 3 0 1 3 0 31 32 (31) 0 4 6 0 4 6 10 EXAMPLE 8. If 2 4 1 11 12 1 3 0 , 21 22 5 4 6 0 1 3 11 12 2 4 1 , 21 22 5 8 2 (37) (38) If m = p, and n = q and each Aij block is of the same form as the corresponding Bij block. 1111 1221 1112 1222 . 2111 2221 2112 2222 11 10.3 The Transpose Matrix Given any m × n matrix aij , we define the transpose of A, denoted as AT and read as “A-transpose”, as a11 a21 a a22 12 T A a ji a1n a2 n am1 am 2 , amn aijT =a ji (1) (2) Theorem 10.3.1 Properties of the Transpose A T T A, A B AT +BT , T A AT , T AB BT AT , T (3a) (3b) (3c) (3d) 12 Proof of (3d): Let AB ≡ C = {cij} n cij aik bkj (4) k 1 n n n cij T c ji a jk bki bki a jk bikT akjT . k 1 C T T T k 1 or ( )T T T (C )T CT T T , Let x1 x xn k 1 (5) and (CD)T DT CT T T , y1 y yn (7) 13 Then the standard dot product n x y x1 y1 x2 y2 ... xn yn x j y j j 1 and in matrix language If x y xT y (8) (9) T We say that A is symmetric, and if T , (10) We say that it is skew-symmetric (or antisymmetric). 14 10.4 Determinants as We denote the determinant of an n × n matrix aij det a11 a12 a1n a21 a22 a2 n an1 an 2 ann (1) The determinant of an n × n matrix aij is defined by n the cofactor expansion det a jk jk , (2) 1 where the summation is carried out on j for any fixed value of k ( 1 k n ) or on k for any fixed value of j ( 1 j n ). Ajk is called the cofactor of the ajk element and is defined as jk (1) j k M jk , (3) where Mjk is called the minor of ajk, namely, the determinant of the (n-1)x(n-1) matrix that survives when the row and the 15 column containing ajk are struck out. Example 1 Find the determinant of the matrix det A a11 (-1)(1+1) M11 +a12 (-1)(1+2) M12 +a13 (-1)(1+3) M13 0 2 -1 A 4 3 5 2 0 -4 a11M11 -a12 M12 +a13 M13 det A a 21 (-1)(2+1) M 21 +a 22 (-1)(2+2) M 22 +a 23 (-1)(2+3) M 23 a 21M 21 +a 22 M 22 -a 23M 23 det A a13 (-1)(1+3) M13 +a 23 (-1)(2+3) M 23 +a 33 (-1)(3+3) M 33 a13M13 -a 23 M 23 +a 33 M33 =(-1) 43 20 -(5) 02 20 +(-4) 02 43 (1)(6) (5)(4) (4)(8) 58 A square matrix aij is upper triangular if aij = 0 for all j < i and lower triangular if aij = 0 for all j > i. If a matrix is upper triangular or lower triangular it is said to be triangular. 16 a11 a12 0 a 22 0 a1n a11 0 a a2 n a22 21 and ann an1 an 2 (upper triangular) 0 0 , ann (lower triangular) Properties of Determinants D1. If any row (or column) of A is modified by adding α times the corresponding elements of another row (or column) to it, yielding a new matrix B, det B = det A. Symbolically: rj → rj+αrk D2. If any two rows (or column) of A are interchanged, yielding a new matrix B, then det B = -det A. Symbolically: rj ↔ rk D3. If A is triangular, then det A is simply the product of the diagonal elements, det A = a11a22…ann 17 D4. If all the elements of any row or column are zero, then det A = 0. D5. If any two rows or columns are proportional to each other, then det A = 0. D6. If any row (column) is a linear combination of other rows (columns), then det A = 0. D7. If all the elements of any row or column are scaled by α, yielding a new matrix B, then det B = αdet A. D8. det (αA) = αn det A. D9. If any one row (or column) a of A is separated as a = b + c, then det |a det |b det |c 18 D10. The determinant of A and its transpose are equal, det( ) det T D11. In general, det( ) det + det D12. The determinant of a product equals the product of their determinants, det( ) (det )(det ) det( ) det + det (11) (12) 19 10.5 Rank; Application to Linear Dependence and to Existence and Uniqueness for Ax = c DEFINITION 10.5.1 Rank A matrix A, not necessarily square, is of rank r, or r(A), if it contains at least one r × r submatrix with nonzero determinant but no square submatrix larger than r × r with nonzero determinant. A matrix is of rank 0 if it is a zero matrix. EXAMPLE 1. Let 2 1 A 0 3 1 4 1 3 5 0 6 9 (1) 20 We may regard the rows of an m × n matrix aij as n-dimensional vectors, which we call the row vectors of A and which we denote as r1, …, rm. Similarly, the columns are m-dimensional vectors, which we call the column vectors of A and which we denote as c1, …, cn. Further, we define the vector spaces span{r1, …, rm} and span {c1, …, cn} as the row and column spaces of A, respectively. The elementary row operations: 1.Addition of a multiple of one row to another Symbolically: rj → rj + αrk 2. Multiplication of a row by a nonzero constant Symbolically: rj → αrj 3. Interchange of two rows Symbolically: rj ↔ rk 21 THEOREM 10.5.1 Elementary Row Operations and Rank Row equivalent matrices have the same rank. That is, elementary row operations do not alter the rank of a matrix. Elementary A row operations B r(A) = r(B) THEOREM 10.5.2 Rank and Linear Dependence For any matrix A, the number of LI row vectors is equal to the number of LI column vectors and these, in turn, equal the rank of A. 22 EXAMPLE 5. Application to Stoichiometry. 1 CO O2 CO2 , 2 1 H 2 O2 H 2O, 2 3 CH 4 O2 CO 2 H 2O, 2 CH 4 2O2 CO2 2 H 2O, 1 CO O2 CO2 0, 2 1 H 2 O2 H 2O 0, 2 3 CH 4 O2 CO 2 H 2O 0, 2 CH 4 2O2 CO2 2 H 2O 0, (4a) (4b) (4c) (4d) (5) 23 CO O2 CO2 H2 H2O CH4 1 0 A 1 0 1 2 1 2 3 2 2 1 0 0 0 1 1 0 0 2 1 0 2 0 0 (6) 1 1 CO O2 1 0 0 0 1 2 1 0 0 CO2 H2 H2O CH4 1 0 0 0 2 2 1 4 2 0 0 0 0 0 1 0 1 CO O2 CO2 0, 2 O2 2 H 2 2 H 2O 0, CO 1 O2 CO2 0, 2 H 2 1 O2 H 2O 0, 2 (7) (8) CO2 4 H 2 2 H 2O CH 4 0, CH 4 3 O2 CO 2 H 2O 0, (5) 2 CH 4 2O2 CO2 2 H 2O 0, 24 Example 6 x1 x2 x3 3 x4 2 x6 4 3x3 3x4 x5 +6 x6 3 x1 2 x1 x2 2 x3 x4 x5 +7 x6 9 5 x3 8 x4 x5 +7 x6 1 x1 1 -1 0 1 0 0 0 0 1 0 0 0 13 0 2 0 -1 25 0 00 0 2 4 4 -1 1 -2 0 0 0 1 0 0 1 -1 0 1 0 0 0 0 6 2 2 - 5 -1 3 1 5 1 0 -1 2 2 0 0 0 0 0 0 -9 1 0 3 -1 9 2 1 1 -1 1 0 Ac 2 -1 1 0 1 3 0 2 4 3 3 -1 6 3 2 1 -1 7 9 5 8 -1 7 1 1 3 0 2 4 1 0 0 1 0 -5 -1 3 -1 0 0 1 5 0 1 -1 2 2 0 0 0 0 0 0 0 1 -2 -1 5 5 0 -5 -1 3 1 1 5 0 1 -1 2 2 0 0 0 0 0 1 5 3 2 2 x2 1- 31 2 5 3 x3 -1- x1 6 - 9 9 1 2 3 2 2 25 Conclusions 9 9 9 9 6 1 2 3 2 2 1 2 2 6 1 1 3 5 1 3 1 2 3 5 1 1 5 0 1 5 3 1 2 3 x 1 1 2 2 0 2 2 0 0 1 3 0 1 0 2 0 0 0 1 0 1 =x0 +1x1 + 2x2 + 3x3 X0 is a particular solution of Ax = c, and x1,x2,x3 are homogeneous solutions. That is, A(x0 +1x1 + 2x2 + 3x3)=c 1 5 3 2 2 x2 1- 31 2 5 3 x3 -1- x1 6 - 9 9 1 2 3 2 2 9 2 3 1 x1 , x 1 , x3 2 0 0 1 2 5 -5 2 1 0 0 1 9 1 0 0 1 0 26 Suppose that a system Ax = c, where A is m x n, has a p-parameter family of solutions x = x0 +1x1 + ..+ pxp Then x0 is necessarily a particular solution, and x1,…, xp, are necessarily LI homogeneous solutions. We call span{x1,…,xp} as the solution space of the homogeneous equation Ax = 0, or the null space of A. The dimension of that null space is called the nullity of A. Theorem 10.5.3 Existence and Uniqueness for Ax = c Consider the linear system Ax = c, where is m x n. There is 1. no solution if and only if r(A|c) ≠r(A) 2. A unique solution if and only if r(A|c) = r(A) = n 3. An (n-r)-parameter family of solutions if and only if r(A|c) = r(A) ≡ r is less than n. 27 Theorem 10.5.4 Homogeneous case where A is m x n If A is m x n, then Ax = 0 1. Is consistent. 2. Admits the trivial solution x = 0. 3. Admits the unique solution x = 0 if and only if, r(A) =n. 4. Admits an (n-r)-parameter family of nontrivial solutions, in addition to the trivial solution, if and only if r(A) ≡ r < n. Theorem 10.5.5 Homogeneous case where A is n x n If A is n x n, then Ax = 0 admits the nontrivial solution, besides the trivial solution x = 0, if and only if det A=0 28 Example 7 Dimensional Analysis Consider a rectangular flat plate in steady motion through undisturbed air as shown in Fig. 1. The object is to conduct an experimental determination of the lift force l generated on the airfoil, that is, experimentally determine the functional dependence of l on the various relevant quantities. A List of the relevant variables is given in Table 1. 29 The Buckingham Pi theorem states that: Given a relation among n parameters of the form g (q1 , q2 , q3 ,...., qn ) 0 then the n parameters may be grouped into n-m independent dimensionless ratios, or n parameters, expressed in a functional form by G(1 , 2 ,...., nm ) 0 or 1 G1 ( 2 , 3 ,...., nm ) The number m is usually, but not always, equal to the minimum number of independent dimensions required to specify the dimensions of all the parameters q1, q2, …and qn. Next, we seek all possible dimensionless products of the form 30 Aa B b cV dV0e f g l h That is, we seek the exponents a, …, h such that ( L)a ( L)b ( M 0 L0T 0 )c ( LT 1 ) d ( LT 1 )e ( ML3 ) f ( ML1T 1 ) g ( MLT 2 ) h M 0 L0T 0 Equating exponents og L,T, M on both sides, we see that a,…,h must satisfy the homogeneous linear system. a b d e3f g h 0 -d -e - g - 2h 0 (26) f g h0 Solving Eq. (26) by Gauss elimination gives the five-parameter family of solutions 31 a 2 1 0 0 1 b 0 0 0 0 1 c 0 0 0 1 0 d 2 1 1 0 0 1 2 3 4 5 e 0 0 1 0 0 f 1 1 0 0 0 g 0 1 0 0 0 h 1 0 0 0 0 (27) where 1,…,a5 are arbitrary constants. With 1 = 1 and 2 = …= 5 = 0, Eq. (27) gives a = -2, b = c = 0, and hence the nondimensional parameter becomes A2 B 0 0V 2V00 1 0l1 that is l/(V2A2). 32 Set 2 = 1, and the other j’s = 0 gives (Reynolds number), Re AV / Set 3 = -1, and the other j’s = 0 gives (Mach number), M V / V0 Set 4 = 1, and the other j’s = 0 gives incident angel, Set 5 = 1, and the other j’s = 0 gives aspect ratio, AR / Therefore, we can conclude that l f (Re, M , , AR) 2 2 A V (28) 33 10.6 Inverse matrix, Cramer’s rule, factorization 10.6.1 Inverse matrix For a system of linear algebraic equations expressed as Ax = c (1) Let us try to find a matrix A-1 having the property that A-1A = I. and A-1Ax = A-1c becomes I x = A-1c since I x = x, we have the solution x = A-1c We call A-1 the inverse of A or A-inverse. 34 A11 A21 A A 1 12 22 1 A det A A1n A2n An1 An 2 Ann (16) The matrix in (16) is called the adjoint of A and is denoted as adjA, so 1 1 A det A adjA (17) If detA≠0, then A-1 exists, in this case we say that A is invertible. If detA = 0, then A-1 does not exist, and we say that A is singular. Example 2 Determine the inverse of 3 2 -1 A 0 1 4 1 5 -2 自行練習 35 3 2 -1 A 0 1 4 , det A 57 0 1 5 -2 T 1 4 0 4 0 1 5 -2 1 -2 1 5 T -22 4 -1 -22 -1 9 2 -1 3 -1 3 2 -1 -5 -13 4 -5 -12 adjA 5 -2 1 -2 1 5 9 -12 3 -1 -13 3 3 2 2 -1 - 3 -1 1 4 0 4 0 1 -22 -1 9 1 1 -1 A = adjA 4 -5 -12 detA 57 -1 -13 3 36 Theorem 10.6.1 Inverse matrix Let A be n x n. if detA ≠0, then there exists a unique matrix A-1such that A-1A = AA-1 = I (27) A is then said to be invertible, and its inverse is given by Eq. (17). If detA = 0, then a matrix A-1 does not exist, and A is said to be singular. Theorem 10.6.2 Solution of Ax = c Let A is n x n and detA ≠0, then Ax = c admits the unique solution x = A-1c. Properties of inverses I1. If A and B are of the same order, and invertible, then AB is too, and (AB)-1 = B-1A-1 (28) 37 I2. If A is invertible, then and (AT)-1 = (A-1)T (29) 1 det(A ) det A 1 (30) I3. If A is invertible, then (A-1)-1 = A and (Am)n = Amn for any integers m and n (positive, negative, and zero). I4. If A is invertible, then AB = AC implies that B = C, BA = CA implies B = C, AB = 0 implies that B = 0, and BA = 0 implies that B = 0. 10.6.3 Cramer’s rule We have seen that if A is n x n and detA≠ 0, then Ax = c has the unique solution x = A-1c (38) 38 Eq. (38) can be expressed as x1 11 12 xn n1 n 2 1n c1 j 1 j c j nn cn j nj c j (39) Equating the ith components on the left with the ith component on the right, we have the scale statement xi ij c j (40) j for any desired i (1 ≦ i ≦ n). Or, recalling Eq. (17) xi ( j Aji det A ) cj j Aji c j det A (41) 39 Theorem 10.6.3 Cramer’s Rule If Ax = c where A is invertible, then each component xi of x may be computed as the ratio of two determinants; the denominator is detA, and the numerator is also the determinant of the A matrix but with the ith column replaced by c. Example 3 Solve the system 5 1 -2 x1 1 -2 0 3 3 1 3 3 1 0 1 5 0 1 -2 1 1 1 1 0 -2 1 , x2 0 8 1 3 0 1 -2 3 1 1 0 1 1 1 3 0 x1 5 -2 3 1 x 1 2 0 1 1 xn -2 1 3 5 -2 3 1 0 1 -2 29 13 , x2 1 3 0 8 8 -2 3 1 0 1 1 (42) (43) 40 10.6.4 Evaluation of A-1 by elementary row operations If we solve a system Ax = c of n equations in n unknowns, or equivalently Ax = Ic, by gauss-Jordan reduction, the result is the form x = A-1c, or equivalently I x = A-1c. 1 3 0 A= -2 3 1 0 1 1 1 3 0 1 0 0 1 3 0 1 0 0 1 0 0 1 4 -3 8 3 8 A I = -2 3 1 0 1 0 0 9 1 2 1 0 0 1 0 1 4 1 8 -1 8 0 1 1 0 0 1 0 1 1 0 0 1 0 0 1 -1 4 -1 8 9 8 1 4 -3 8 3 8 A-1 = 1 4 1 8 -1 8 -1 4 -1 8 9 8 41 10.6.5 LU-factorization LU-factorization is an alternative method of solution that is based upon the factorization of an n x n matrix A as a lower triangular matrix L times an upper triangular matrix U: A=LU= 11 21 31 0 0 u11 u12 u13 0 u u 0 22 22 23 0 u 33 32 33 0 If we carry out the multiplication on the right and equate the nine elements of LU to the corresponding elements of A we obtain nine equations in the 12 unknown lij’s and uij’s. Since we have more unknowns than equations, there is some flexibility in implementing the idea. According to Doolittle’s Method we can set each lii = 1 in L and solve uniquely for remaining lij’s and uij’s. 42 With L and U determined, we then solve Ax = LUx = c by setting Ux = y so that L(Ux) = c breaks into the two problems Ly = c Ux = y (49a) (49b) each of which is simple because L and U are triangular. We solve Eq. (49a) for y, put that y into Eq. (49b), and then solve Eq. (49b) for x. 2 -3 3 x1 -2 Example 5 Solve by the Doolittle 6 -8 7 x2 = -3 -2 6 -1 x3 3 LU-factorization method. 2 -3 3 1 0 0 u11 u12 u13 6 -8 7 = 21 1 0 0 u22 u23 -2 6 -1 31 32 1 0 0 u 33 43 2 -3 3 u11 6 -8 7 = u 21 11 -2 6 -1 31u11 u12 u +u22 21 12 u + 31 12 u 32 22 u13 u + u 21 13 23 31u13 + 32 u23 +u 33 In turn: u11 = 2 u12 = -3 u13 3 21 6 / u11 3 u22 8 u23 7 u 1 21 12 u 7 (3)(3) 2 21 13 31 2 / u11 1 32 (6 31 12 u33 1 31 13 u )/u22 3 u u =-1-(-1)(3)-(3)(2)=8 32 23 44 Then Eq. (49a) becomes 1 0 0 y1 2 3 1 0 y 3 2 1 3 1 y3 3 which gives y = [-2, 3, -8]T. Finally Eq. (49a) becomes 2 -3 3 x1 -2 0 1 -2 x 3 2 0 0 8 x3 -8 which gives the final solution x = [2, 1, -1]T. 45 10.7 Change of Basis Let B= {e1,…,en} be a given basis for the vector space V under consideration so that any vector x in V can be expanded as (1) x = x1e1 +…+ xnen If we switch to some other basis B’= {e’1,…,e’n}, then we may expand the same vector x as x = x’1e’1 +…+ x’ne’n (2) We may expand each of the ej’s in terms of B’: e1 q11e1 qn1en (3) en q1n e1 qnn en 46 Putting Eq. (3) into (1) gives x x1 (q11e1 ... qn1en ) ... xn (q1n e1 ... qnn en ) ( x1q11 ... xn q1n )e1 ) ... ( x1qn1 ... xn qnn )en (4) a comparison of Eq. (2) and (4) gives the desired relations x1 q11 x1 ... q1n xn (5) xn qn1 x1 ... qnn xn or, in matrix notation where q11 Q qn1 q1n qnn [x]B’ = Q [x]B (6) and (7) x1 x1 [x]B , [x]B xn xn (8) 47 We call [x]B the coordinate vector of the vector x with respect to the ordered basis B, and similarly for [x]B’, and we call Q the coordinate transformation matrix from B to B’. In the remainder of this section we assume that both bases, B and B’, are ON. Thus, let us rewrite Eq. (3) as eˆ1 q11eˆ1 qn1eˆ n (9) eˆ n q1n eˆ1 qnn eˆ n If we dot ê1 into both sides of the first equation in Eq. (9), we obtain q11 = eˆ1 eˆ1 . Dotting ê2 gives q21 = eˆ 2 eˆ1 . Dotting ên gives qn1 = eˆ n eˆ1. the result is in the formula qij eˆ i eˆ j (10) 48 which tell us how to compute the transformation matrix Q. Two properties of Q, the first of these is Q-1 = QT q11 q 12 T Q Q q1n so that qn1 q11 q12 qn 2 qn1 qn 2 qnn Q-1 = QT q1n eˆ1 eˆ1 qnn eˆ n eˆ1 eˆ1 eˆ n I eˆ n eˆ n (11) (12) The second of these is detQ = ±1 QTQ = implies that det(QTQ) = det = 1. But det(QTQ) = (detQT)(detQ) = (detQ)(detQ) = (detQ)2. Hence detQ must be +1 or -1. 49 Example 1 Consider the vector space R2, with the ON bases B eˆ1 , eˆ2 and B eˆ1, eˆ2 . B’ is obtained from B by a counterclockwise rotation through an angle . From the figure, we have q11 eˆ1 eˆ1 (1)(1) cos cos q12 eˆ1 eˆ2 (1)(1) cos( ) sin 2 q21 eˆ2 eˆ1 (1)(1) cos( ) sin 2 q11 eˆ2 eˆ2 (1)(1) cos cos so that the coordinate transformation cos sin matrix is Q sin cos Hence x1 cos sin x1 x sin cos x 2 2 50