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Solving Simultaneous Equations (An introduction to Linear Algebra) Systems of Equations • Many geoscience problems require using systems of equations. • A common application for systems of equations involves estimating the fraction of several components in a mixture. • Examples: – Estimating the mineralogical composition of a rock based on chemical analyses – Deriving source rocks contribution to glacial till based on the chemistry of various size fractions. Model Compositions • A calculated source composition of till or the mineralogical composition of a rock based on its major oxide chemistry is a modeled composition. How does it work? • Let’s assume that each element (or chemical species) exists in the same proportion in each component. • The way each element is distributed can be expressed linearly: Where the coefficient a is the ij a11x1+a12x2+…+a1mxm=b1 known content of element i in component j. b is the total a21x1+a22x2+…+a2mxm=b2 content of element i in the mixture. … For m unknown components, an1x1+an2x2+…+anmxm=bn the proportions xj to be solved are the same for n elements. an1x1+an2x2+…+anmxm=bn What we know: a b All we need is xm If m=n then we have the same number of equations as we have of unknowns! This is a simple system of linear equations. Example (from Mäkinen and Gustavsson, 1998) • A stone count near glacial till in Finland roughly contains 20% granitoids, 53% quartzite, and 20% amphibolite. • This does not agree with the chemical composition of till corresponding to these 3 rock types. • A layer of gravel sand is found beneath the till. Based on appearance, it is thought that the gravel-sand layer is where the till originated. Setting up the model • Include the following chemical species which exist in known amounts in each of the components: • Al2O3, CaO, MgO, Cr, Ni, Pb, Rb, Sr, V, Y, Na2O, SiO2, TiO2, K2O • Have bulk analysis of the 4 components: granitoids (A), quartzite(B), amphibolite(C), gravel sand(D). • Have bulk analysis of the till (T). Let’s take the example of Al2O3: AAl2O3xA + BAl2O3xB+ CAl2O3xC + DAl2O3xD = TAl2O3 ACaOxA + BCaOxB+ CCaOxC + DCaOxD = TCaO … AK2OxA + BK2OxB+ CK2OxC + DK2OxD = TK2O Solution • Check to see that it adds up! xA = 0.26, xB = 0.03, xC = 0.09, xD = 0.62 • Notice that the equations are linear, of the form Ax = B. • Need to find an easy way to multiple equations simultaneously. • We can represent the equations in the last example as coefficients and constants, omitting the variables. AAl2O3+ BAl2O3+ CAl2O3 + DAl2O3 = TAl2O3 xA TAl2O3 ACaO + BCaO+ CCaO+ DCaO = TCaO xB = TCaO … xC … AK2O + BK2O+ CK2O + DK2O = TK2O xD TK2O A x = B Matrix Algebra • Matrices provide a theoretical and practical way of solving problems. • Excel is a natural medium by which to construct, analyze and solve matrices. What is a Matrix? Monthly Expenses Food Utilities Health Family A a b c Family B d e f What is a Matrix? Matrices Size of a Matrix • Defined by number of ROWS and number of COLUMNS. • In this case, the matrix on the left has 2 rows and 3 columns so it is a (2x3) matrix. • More generally, it has m rows and n columns so we say it’s an (mxn) matrix. Matrices The elements of a matrix can be variables or functions instead. It provides a grid arrangement for data, much like the way Excel manages data. Matrices can be added, subtracted, multiplied and, in general, manipulated to extract information. Examples of matrices • A matrix that has equal number of rows and columns, i.e., m=n is called a square matrix. • A matrix with 4 rows and 2 columns is a 4x2 matrix. • How would you describe the matrix to the right in terms of mxn? Matrix Operations • To add or subtract two matrices, simply add the elements together. Matrices must be the same size (i.e., same mxn) • Or more generally: Matrix Operations • As implied by the last example, you can apply an operation to an entire matrix. Adding a matrix to itself is the same as doubling a matrix. Therefore we can multiply a matrix by a constant. Matrix Operations • What about multiplying two matrices? • Can only be done if the number of rows in matrix A equals the number of columns in matrix B. • For example: How to multiply matrices? Or In other words if matrix A is an mxn matrix and matrix B is a kxl matrix, n must equal k in order to multiply them. The result will be an mxl matrix. However AB ≠ BA! More Properties: Adding and Multiplying Matrices • If A, B, and C are matrices and a and b are numbers, (A+B)C = AC + BC A(B+C) = AB + AC A(A+B) = aA + bB (a+b)A = aA + bA Example The Identity Matrix If we have the matrices It’s easy to see that And The Identity Matrix We call these two forms linear combinations. From this we can see how a matrix can behave like the number 1 behaves in normal multiplication. The Identity Matrix A matrix that behaves like the number 1. For any nxn matrix A, AIn = InA = A Invertible Matrices An nxn matrix A is invertible if AB = BA = In Where In is the Identity Matrix. If this is the case, B is the inverse of A. Check to see if this is true: The notation for “inverse of matrix A” is A-1 For a matrix to be invertible, it must be square. Using Invertible Matrices • How do we use them? Remember that we want to solve systems of equations of the form Ax = B • Multiply each side by the inverse of A. A-1(AX) = A-1(B) • Regroup the factors (A-1A)X = A-1B • Regroup the factors, and we know that A-1A= I X = A-1B Example 3x + 2y – 5z = 12 x – 3y + 2z = -13 5x – y + 4z = 10 Coefficients matrix Variables matrix Constants matrix 3 2 -5 x 12 1 -3 2 y -13 5 -1 4 z 10 Excel Commands You will need to know a)How to transpose matrix A. Use the =TRANSPOSE(A) a)how to multiply matrix A-1 by matrix B – use the =MMULT(range of cells from matrix A-1, range of cells for matrix B) function. Make sure your matrices are compatible for multiplication. b) how to invert a matrix A; use the function =MINVERSE(A) As with any Excel formula, you may combine various operations in one cell. Because you are selecting multiple cells for input and output, make sure you hold the CTRL-SHIFT keys when pressing ENTER.

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linear equations, linear algebra, simultaneous equations, how to, systems of equations, systems of linear equations, elementary algebra, solving equations, word problems, quadratic equations, system of equations, system of linear equations, real numbers, abstract algebra, o zero

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posted: | 1/19/2010 |

language: | English |

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