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If r < n, there will be more than one solution (infinitely many if the field is infinite). For all solutions are obtained by taking the unknowns xc1 , · · · , xcr as dependent unknowns and using the r equations corresponding to the non–zero rows of B to express these unknowns in terms of the remaining independent unknowns xcr+1 , . . . , xcn , which can take on arbitrary values: x c1 x cr = b1 n+1 − b1cr+1 xcr+1 − · · · − b1cn xcn . . . = br n+1 − brcr+1 xcr+1 − · · · − brcn xcn ....
The last matrix is in reduced row–echelon form. To solve the system of equations whose augmented matrix is the given matrix over Z3 , we see from the reduced row–echelon form that x = 1 and y = 2 − 2z = 2 + z , where z = 0, 1, 2. Hence there are three solutions to the given system of linear equations: (x, y , z ) = (1, 2, 0), (1, 0, 1) and (1, 1, 2)....
5. Find the value of t for which the following system is consistent and solve the system for this value of t. x+y = 1 tx + y = t (1 + t)x + 2y = 3. [Answer: t = 2; x = 1, y = 0.]...
is singular. For it can be verified that A has reduced row–echelon form 101 0 1 1 000...
Hence X t X = −X t X and X t X = 0. But if X = [x1 , . . . , xn ]t , then X t X = 2 x2 + . . . + xn = 0 and hence x1 = 0, . . . , xn = 0. 1...
3.3. LINEAR DEPENDENCE REMARK 3.3.1 If X1 , . . . , Xm are linearly independent and x 1 X1 + · · · + x m Xm = y 1 X1 + · · · + y m Xm , then x1 = y1 , . . . , xm = ym . For the equation can be rewritten as (x1 − y1 )X1 + · · · + (xm − ym )Xm = 0 and so x1 − y1 = 0, . . . , xm − ym = 0....
where not all of x1 , . . . , xm+1 are zero. Now if xm+1 = 0, we would have x1 X1 + · · · + xm Xm = 0, with not all of x1 , . . . , xm zero, contradictiong the assumption that X1 . . . , Xm are linearly independent. Hence xm+1 = 0 and we can use equation 3.1 to express X as a linear combination of X1 , . . . , Xm : X= −x1 −xm X1 + · · · + Xm . xm+1 xm+1...
73 expansion: det A = a11 M11 (A) − a12 M12 (A) + . . . + (−1)1+n M1n (A) jn = (−1)1+j a1j M1j (A).
=1...
Find the values of k for which det A = 0 and hence, or otherwise, determine the value of k for which the following system has more than one solution: 2x + 3y + k z = 3 x+y−z = 1...
Calculating with complex numb ers
We can now do all the standard linear algebra calculations over the field of complex numbers – find the reduced row–echelon form of an matrix whose elements are complex numbers, solve systems of linear equations, find inverses and calculate determinants. For example, = 1 +i 2−i (1 + i)(8 − 2i) − 7(2 − i) 7 8 − 2i = (8 − 2i) + i(8 − 2i) − 14 + 7i = −4 + 13i = 0...
De Moivre’s theorem
The next theorem has many uses and is a special case of theorem 5.6.3(ii). Alternatively it can be proved directly by induction on n. THEOREM 5.7.1 (De Moivre) If n is a positive integer, then (cos θ + i sin θ)n = cos nθ + i sin nθ. As a first application, we consider the equation z n = 1. THEOREM 5.7.2 The equation z n = 1 has n distinct solutions, namely 2kπ i the complex numbers ζk = e n , k = 0, 1, . . . , n − 1. These lie equally spaced on the unit circle |z | = 1 and are obtained by starting at 1, moving π round the circle anti–clockwise, incrementing the argument in steps of 2n . (See Figure 5.7) 2π i We notice that the roots are the powers of the special root ζ = e n ...
Now AB 2 = (x2 − x1 )2 + (y2 − y1 )2 , OA2 = x2 + y1 , OB 2 = x2 + y2 . 1
(7.3)
Substituting in equation 7.3 then gives
2 2 (x2 − x1 )2 + (y2 − y1 )2 = (x2 + y1 ) + (x2 + y2 ) − 2OA · OB cos θ, 1 2
132
CHAPTER 7. IDENTIFYING SECOND DEGREE EQUATIONS
which simplifies to give OA · OB cos θ = x1 x2 + y1 y2 = X · Y . It follows from theorem 7.1.2 that if A = (xx, y1 )aand B = (x2 , y,2 ) are 1 x 2 1 then nd Y = points distinct from O = (0, 0) and X = y2 y1 X · Y = 0 means that the rays OA and OB are perpendicular. This is the reason for the following definition: DEFINITION 7.1.3 (Orthogonal vectors) Vectors X and Y are called orthogonal if X · Y = 0. There is also a connection with orthogonal matrices: THEOREM 7.1.3 Let P be a 2 × 2 real matrix. Then P is an orthogonal matrix if and only if the columns of P are orthogonal and have unit length. Pro of. P is orthogonal if and only if P t P = I2 . Now if P = [X1 |X2 ], the matrix P t P is an important matrix called the Gram matrix of the column vectors X1 and X2 . It is easy to prove that X . 1 · X 1 X1 · X 2 t P P = [Xi · Xj ] = X2 · X 1 X2 · X 2 Hence the equation P t P = I2 is equivalent to =1 X 0 1 · X 1 X1 · X 2 01 X2 · X 1 X2 · X 2 or, equating corresponding elements of both sides: X1 · X1 = 1, X1 · X2 = 0, X2 · X2 = 1, which says that the columns of P are orthogonal and of unit length. The next theorem describes a fundamental property of real symmetric matrices and the proof generalizes to symmetric matrices of any size. THEOREM 7.1.4 If X1 and X2 are eigenvectors corresponding to distinct eigenvalues λ1 and λ2 of a real symmetric matrix A, then X1 and X2 are orthogonal vectors...
EXAMPLE 7.1.1 Let A be the symmetric matrix 1 . 2 −6 A= −6 7 Find a proper orthogonal matrix P such that P t AP is diagonal. Solution. The characteristic equation of A is λ2 − 19λ + 48 = 0, or (λ − 16)(λ − 3) = 0. Hence A has distinct eigenvalues λ1 = 16 and λ2 = 3. We find corresponding eigenvectors a 2. − 3 nd X2 = X1 = 3 2 √ Now ||X1 || = ||X2 || = 13. So we take 1 X1 = √ 13 − 3 2 a 1 nd X2 = √ 13 3 2 ....
(iii) AB + B C =AC (the triangle law); (iv) B C =AC − AB = C − B; (v) if X is a vector and A a point, there is exactly one point B such that AB = X , namely that defined by B = A + X .
E E E...
Solution. The required plane has the form λ(x + y − 2z − 1) + µ(x + 3y − z − 4) = 0, where not both of λ and µ are zero. Substituting the coordinates of P 0 into this equation gives −2λ + µ(−4) = 0, So the required equation is −2µ(x + y − 2z − 1) + µ(x + 3y − z − 4) = 0, or −x + y + 3z − 2 = 0. Our final result is a formula for the distance from a point to a plane. λ = −2µ....
19. The points A = (1, 1, 5), B = (2, 2, 1), C = (1, −2, 2) and D = (−2, 1, 2) are the vertices of a tetrahedron. Find the equation of the line through A perpendicular to the face B C D and the distance of A from this face. Also find the shortest distance between the skew lines AD and B C . √ [Ans: P = (1 + t)(i + j + 5k); 2 3; 3.]...
[12] J.M. Geramita and N.J. Pullman. An Introduction to the Application of Nonnegative Matrices to Biological Systems, 1984. Queen’s Papers in Pure and Applied Mathematics 68. Queen’s University, Kingston, Canada. [13] M. Pearl. Matrix Theory and Finite Mathematics, 1973. McGraw–Hill Book Company, New York. [14] J.G. Kemeny and J.L. Snell. Finite Markov Chains, 1967. Van Nostrand Reinhold, N.J. [15] E.R. Berlekamp. Algebraic Coding Theory, 1968. McGraw–Hill Book Company, New York. [16] G. Strang. Linear Algebra and its Applications, 1988. Harcourt Brace Jovanovich, San Diego. [17] H. Minc. Nonnegative Matrices, 1988. John Wiley and Sons, New York. [18] G.C. Preston and A.R. Lovaglia. Modern Analytic Geometry, 1971. Harper and Row, New York. [19] J.A. Murtha and E.R. Willard. Linear Algebra and Geometry, 1969. Holt, Rinehart and Winston, Inc. New York. [20] L.A. Pipes. Matrix Methods for Engineering, 1963. Prentice–Hall, Inc. N. J. [21] D. Gans. Transformations and Geometries, 1969. Appleton–Century– Crofts, New York. [22] J.N. Kapur. Transformation Geometry, 1976. Affiliated East–West Press, New Delhi. [23] G.C. Reid. Postscript Language Tutorial and Cookbook, 1988. Addison– Wesley Publishing Company, New York. [24] D. Hearn and M.P. Baker. Computer Graphics, 1989. Prentice–Hall, Inc. N. J. [25] C.G. Cullen. Linear Algebra with Applications, 1988. Scott, Foresman and Company, Glenview, Illinois. [26] R.E. Larson and B.H. Edwards. Elementary Linear Algebra, 1988. D.C. Heath and Company, Lexington, Massachusetts Toronto. 192...
inversion 76 invertible matrix 37 Joachimsthal 168 least squares normal equations 48 residuals 48 linear combination 17 linear equation 1 linear transformation 27 reflection 29 rotation 28 pro jection 30 linear independence 41, 60 left–to–right test lower half plane 98 Markov matrix 54 mathematical induction 32 matrix 23 addition 23 additive inverse 24 equality 23 inverse 37 orthogonal 136 power 32 product 25 proper orthogonal 136 rank 68 scalar multiple 24 singular 37 skew–symmetric 47 symmetric 47 subtraction 24 transpose 46 minor 74 modular addition 4 multiplication 4 non–singular matrix 37, 50 non–trivial solution 17 orthogonal matrix 120, 136 195...
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