旼컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� � Mat-X V4.1 VECTORS, LINEAR ALGEBRA AND MORE. � 읕컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� SYSTEM REQUIREMENTS: 컴컴컴컴컴컴컴컴컴컴 For running Mat-X V4.1 there are no special hardware requirements. A colour monitor will often be useful though. If you wish to use the graphical func- tions of Mat-X V4.1, you need at least a 256 KB VGA card. NOTE: Some of the words used in Mat-X is directly translated from Danish, and might not be the correct mathematical expressions. SHAREWARE: 컴컴컴컴� Mat-X is a shareware-program. This means that the program can be copied free- ly, and be tested by anyone. If you then after a reasonable try-out time (3-4 weeks), still wish to use the program, you have to register it. Prices: 100 Dkr. or 20 US$. Send the money to this address: �袴袴袴袴袴袴袴� � Address: � 픔컴컴컴컴컴컴캘 튡urple � 튛indevej 2 � 튙rlunde � �3550 Slangerup� 튒enmark, DK � 훤袴袴袴袴袴袴暠 Remember to write your name and address on the letter, so that we can send you a registration card, with your personal ID#. If you find any errors (including wrong use of words in the english version), please write, and tell us about the errors you located. We will then try to correct them, in the future versions of Mat-X. You can write to the above address, or send E-mail to one of our internet addresses: purple@diku.dk mystical@inet.uni-c.dk MANUAL: 컴컴컴 In the elaboration of the program, user-friendliness has been important. It is though made in a way, which requires that the user has a certain know- ledge of the functions used. 旼컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� � The subsequent goes through all of the functions in Mat-X V4.1: � 읕컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� 1. 2D - Vectors: 컴컴컴컴컴컴컴컴 When drawing a vector, all scaling of the axes is decided in the light of the vector, which strecthes furthest out on that axis. This means that this vector strecth to the edge of the axis. 1.1. 2D - Angle between vectors: 컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴 This function calculates one of the angles, between two vectors in the plane. The other angle can be calculates as: (angle2 = 360 - angle1). As input the x and y co-ordinates of the two vectors, are used in the normal co-ordinate system (in linear algebra known as the natural base or the ep- silon base). 1.2. 2D - Projection of vector onto vector. 컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� This function calculates the projection of one vektor onto another. As input it uses the co-ordinates of the vector to project, and then the vector to project onto. In the graphic for this function, the two starting vectors is drawn yellow, and the projection is drawn cyan. 1.3. 2D - Area of parallelogram. 컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴 The function calculates the area of a parallelogram, with is stretched out between two vectors. As input the co-ordinates of the two vectors are used. 2. 3D - Vectors. 컴컴컴컴컴컴컴컴 When drawing vectors in 3D, the axis are scaledusing the same rules as with 2D vectors. In the drawings the result vector is always drawn in cyan, while the vectors which are part of the calculation are drawn in yellow. It is possible to rotate the co-ordinate system using the cursor keys. 2.1. 3D - Length of vector. 컴컴컴컴컴컴컴컴컴컴컴컴컴� Calculates the length of a vector. The input is the x, y and z co-ordinates of the vector. 2.2. 3D - Projection of vector onto vector. 컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� This function calculates the vector which is the projection of a vector, onto another vector. The input is the co-ordinates of the vector to project, and then the vector to project onto. 2.3. 3D - Projection of vector onto plane. 컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴 Calculates the vector which is the projection of a vector, onto a plane. The input is the co-ordinates of the vector, and a normal vector for the plane (see mathematical reference). 2.4. 3D - Vector product. 컴컴컴컴컴컴컴컴컴컴컴컴� Calculates the product of two vectors, using the two vectors co-ordinates. 2.5. 3D - Distance from point to plane. 컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� Calculates the distance from a point to a plane (the distance is the length of the vector which starts at the point, and is perpendicular on the plane). The input is the co-ordinates of the point and a, b, c and d, in an equation for the plane. The equation must have this syntax: (aX + bY + cZ + d = 0). 2.6. 3D - Distance between skew lines. 컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴 The function calculates the perpendicular distance, between to skew lines in 3D space (the distance is the length of the vector, which is perpendi- cular to both lines). As input it uses the co-ordinates of a point on each of the lines, and a vector of direction for each of the lines. 3. Linear algebra. 컴컴컴컴컴컴컴컴컴 If you by a mistake choose a wrong function from the menu, you can press the key (0), when you are asked to choose a memory field for a matrix. This will get you back to the menu. If the result of a calculation is a matrix, you will be asked if you wish to save the matrix. If you answer yes (Y), you can save the result in one of the memory fields A, B, C, ..., S. 3.1. Enter matrixes. 컴컴컴컴컴컴컴컴컴컴 Before you can perform any calculations in the "linear algebra part" of Mat-X, you must type in one or more matrixes. The matrixes are referred to as a letter. You can use the letters A, B, C, ..., S, and only one matrix can be saved as each letter. This means that the maximum number of matrixes you can use at the same time is 19. When you no longer need a matrix, you can save a new one in its memory field. When you choose "Enter matrixes" from the menu, you can see, at the top of the screen, which matrixes that has not yet been used. Now choose the memory letter you wish to save the matrix in. You will now be asked to state the number of rows and coloumns in the matrix. The maximum is 20 coloumns and 20 rows. This should be enough for most calculations. Then you can type in the numbers, which the matrix contains, one by one. The numbers must be entered from top to bottom in the first coloumn, then in the second coloumn etc. 3.2. See matrixes. 컴컴컴컴컴컴컴컴컴 When you have entered a number of matrixes, you have them shoved on the screen by choosing this function from the menu. First the matrix with the lowest letter will be shown. If you wish to see another matrix then press its letter. If a matrix is wider than the screen, you can move left and right in the matrix by using the cursor keys. 3.3. Addition, subtraction og multiplication. 컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� 3.3.1. Add two matrixes. 컴컴컴컴컴컴컴컴컴컴컴컴 The two matrixes you wish to add, must be the same size (this means that the number of rows and coloums, must be the same number in both matrixes). When you add matrixes, the numbers which have the same position in each matrix is added. Example: � 1 3 5� � 4 2 3� � 4+1 3+2 5+3� � 5 5 8� � 2 4 1� + � 1 3 3� = � 1+1 4+3 5+3� = � 2 7 8� � 3 4 4� � 1 2 2� � 3+1 4+2 5+2� � 4 6 7� The input for this function, is the letter for each of the matrixes. 3.3.2. Subtract one matrix from another. 컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴 The two matrixes which are part of the calculation, must have the same size. When subtracting matrixes the second matrix is subtracted from the first matrix, in the same way as when adding (see 4.3.1). The input for the function, is the letter for the start matrix, followed by the matrix to subtract. 3.3.3. Multiply a matrix by a number. 컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� When a matrix is multiplied by a number, all the fields in the matrix is multiplied with that number. The input is the letter of the matrix, and then the number to multiply it with. 3.3.4. Multiply a matrix by a matrix. 컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� The first matrix must have the same number of coloumns, as the second ma- trix has rows. The multiplication of two matrixes is done by multiplying the numbers in the first matrixs' row, with the numbers in the second ma- trix's coloumn, and then adding these numbers. Example: � 1 3 5� � 4 2 � � 1*4+3*1+5*1 1*2+3*3+5*2� � 12 21� � 2 4 1� * � 1 3 � = � 2*4+4*1+1*1 2*2+4*3+1*2� = � 13 18� � 3 4 4� � 1 2 � � 3*4+4*1+4*1 3*2+4*3+4*2� � 20 26� The input for the function, is the letter of the two matrixes to multiply. Remember that (A*B) is different from (B*A). 3.4. Determinant. 컴컴컴컴컴컴컴컴� This function calculates the determinant of a matrix. The matrix must be square. The input for the function, is the letter of the matrix. The function uses a recursive formula for calculating the determinant, and large matrixes (10x10 or higher), the calculation will take some time. If a random 10x10 matrix takes 1 min, a random 11x11 matrix will take 11 minutes. If a matrix contains many 0's, the calculation will not take as much time, as when calculating a matrix with few 0's. 3.5. Convert to echelon. 컴컴컴컴컴컴컴컴컴컴컴컴 This function converts a matrix to an echelon-matrix. This is a matrix which looks like this: �0...0 1 *...* 0 *...* 0 *...*� �0.....0.....0 1 *...* 0 *...*� �. . 0 . .� �. . . . .� �. . . 0 .� �0.....0.......0.....0 1 *...*� �0.....0.......0.......0.....0� �. . . . .� �. . . . .� �0.....0.......0.......0.....0� The input is, the letter of the matrix to convert. 3.6. Find inverse matrix. 컴컴컴컴컴컴컴컴컴컴컴컴� This function calculates the inverse matrix of a random square, and lineary independent matrix. The function does not check if the matrix you choose is lineary independent, you have to check that yourself (see invertibility in mathematical references). The input for the function is, the letter of the matrix to invert. 3.7. Base for the matrixs' core. 컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴 The function calculates a base for the core, of a random matrix. The input is the letter of the matrix, whose core you wish to calculate a base for. 3.8. Ortonormalisation. 컴컴컴컴컴컴컴컴컴컴컴� The function converts a matrix to another matrix, where the coloumns are ortonormal. The ortonormalised matrix has the same base, as the original. The function uses an algorithm by J. P. Gram and E. Schmidt. The input is the letter of the matrix, you wish to ortonormalise. 4. Quadratic equation. 컴컴컴컴컴컴컴컴컴컴컴 The quadratic equation must have this syntax: (aX� + bX + c = 0). Remember that (a) must be different from 0. The input for the function, is the coefficients a, b and c. The result is the solutions for the quadratic equation. The function only calculates the real solutions, not the complex ones. This means that there can be 0, 1 or 2 solutions for the equation. 5. Lines intersection. 컴컴컴컴컴컴컴컴컴컴컴 The function calculates the point of intersection, between two lines in the plane. The lines must have this syntax: (aX + bY = c). The input for the function is, the coefficients a, b and c for both lines. The result is the x and y co-ordinates for the point of intersection. If the two lines are parallel, this will be reported. 6. Roots in a polynomial. 컴컴컴컴컴컴컴컴컴컴컴컴� This function calculates the roots in a polynomial, of a degree between 3 and 20, accurate to the fourth decimal. n n-1 The Polynomial must have the syntax: (aX + bX + ... + cX + d). Where (n) is the degree of the polynomial. The input is, first the degree of the polynomial. Then you enter the coeffi- cients for each term, and last the constant. Now enter the start and end of the interval, where you want the program to search for roots. Since most roots are normally located around the y-axis, an interval from -10 to 10 is often a good choice. The function then searches for roots accurate to the first decimal. It then zoomes in on the roots, by running through the interval around the roots with higher and higher accuracy. The method is not perfect, especially not on a computer where the inaccu- racy on floating point numbers can present problems. Therefore you should always check the results you get from this function. 7. Computation of interest. 컴컴컴컴컴컴컴컴컴컴컴컴컴� 7.1. Extrapolation of capital. 컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴 This function calculates what an amount of money can be increased to, at a certain interest level, if you add the interest after each term. The first input is the start capital (meaning the capital you wish to de- posit on i.e. a bank account), then the interest per term, and then the num- ber of terms you wish to have the money on the account. The result is the capital on the account, after that number of terms. 7.2. Annuity savings. 컴컴컴컴컴컴컴컴컴컴� On an annuity saving, a fixed amount of money is added each term. The input is the deposit per term, the interest rate in % (interest is added after each term), and then the number of terms. The result is the capital on the account, after that number of terms. 7.3. Annuity loan. 컴컴컴컴컴컴컴컴컴 An annuity loan is paid back in small amounts each term. Interest is added after each term. The input is, first the principal (the amount of money you loan), then the interest rate per term, and then the number of terms to pay back the loan. The result is the amount to pay each term. 8. Computtion of probability. 컴컴컴컴컴컴컴컴컴컴컴컴컴컴� Calculates average, variation and dispersion. The input is, first the number of outcomes. Then you enter the value and probability for each outcome. The probabilities are in percent, and the to- tal probabilities must therefor be 100%. The result is the average, variation and dispersion. 9. Distance from point to line. 컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� Calculates the distance from a point, to a line in the plane. The distance is the length of the vector, which starts at the point, and is perpendicular to the line. The line must have this syntax: (Y = aX + b). The input is (a) and (b), in the lines equation, and the x and y co-ordi- nates for the point. 10. Area, volume and circumference. 컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� This function can calculate: Area of: Parallelogram. Trapezium. Ellipse. Area and circumference of circle. Curved surface and volume of: Cylinder. Cone. Volume and surface of sphere. Choose the geometric figure you wish, and enter the needed values. 11. Integral & differential quotient for polynomial. 컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴 Calculates the indeterminate integral and the differential quotient, for a polynomial of a degree between 2 and 20. Furthermore the determinate inte- gral is calculated for an interval. The input is first the degree of the polynomial. Then you enter the coeffi- cients for each term, with the constant as the last. State (0) if the term does not exist. Then the program asks for the start and end of the interval, in which to calculate the determinate integral. The function can also draw the polynomial. When drawing, 124 calculations is used, which should be enough to give a detailed picture of polynomias of even high degrees. The area of the poly- nomial which is drawn, is the area used in the calculation of determined integral. 12. Distance from point to point + the lines equation. 컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴 Calculates the distance between two points in the plane, and the equation of the line which goes through both points. The input is the x and y co-ordinates of the two points. MATHEMATICAL REFERENCS: 컴컴컴컴컴컴컴컴컴컴컴� The mathematical referencs, is ment as a help when solving math problems. Not all problems can be solved just by using formulas. Some need to be re- written, or specific elements must be selected, and used in a formula. The mathematical referencs can therefore be used, as a work of reference. The reference to the right of 'Use:' is of the function i Mat-X V4.1, which can be used for solving the problem. A few of the calculatons are not supported by Mat-X V4.1. Vectors in 3D. 컴컴컴컴컴컴컴 NORMAL VECTOR FOR THE PLANE: - If the planes equation is known, the normal vector is the coefficients a, b and c: (aX + bY + cZ + d = 0). - If the equation is not known, the normal vector is the vector product of two vectors in a plane. Use: "3D vectors - Vector product". THE PLANES EQUATION: - Requires a normal vector (a,b,c) and a point (x0,y0,z0) in a plane Equation: (a(x-x0) + b(y-y0) + c(z-z0) = 0). PROJECTION OF LINE ONTO PLANE: - The same as projecting the lines vector of direction, onto the plane. Use: "2D vectors - Projection of vector on vector". ANGLE BETWEEN PLANES: - The same as the angle between the two planes normal vectors. Use: "3D vectors - Angle between vectors". ANGLE BETWEEN LINE AND PLANE: - The same as the angle between the planes normal vector, and the lines vector of direction. Brug: "3D vectors - Angle between vectors". POINT OF INTERSECTION FOR LINE AND PLANE: - Is found by inserting the lines parameter representation in the planes equation. The t is isolated and the found t-value is inserted in the lines parameter representation whereby the point of intersection can be calculated Example: Line: (x,y,z) = (1+2t, 3+t, -2+4t) Plane: 2X - 4Y + 3Z - 7 = 0 The line is inserted in the planes equation: 2(1+2t) - 4(3+t) + 3(-2+4t) - 7 = 0 t=23/12 The t-value is inserted in the lines parameter representation: (x,y,z) = (1+2*23/12, 3+23/12, -2+4*23/12) = (58/12, 59/12, 68/12) Linear algebra. 컴컴컴컴컴컴컴� INVERTIBILITY: - If a matrix is invertible, it can be checked in two ways: 1) If the determinant is different from 0, then the matrix is invertible. Use: "Linear algebra - Determinant". 2) If the matrix is square and lineary independent, then the matrix is in- vertible. The matrix is lineary independent, if it has initial ones in every coloumn. Use: "Linear algebra - Convert to echelon". RANK OF MATRIXES: - The rank of matrix, can be calculated by converting the matrix, to an echelon matrix. The rank is the number of coloumns with initial ones. Use: "Linear algebra - Convert to echelon". This manual is written by: BRIAN JENSEN 컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴캑 The End 쳐컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴�