(But, they are entirely separate branches of mathematics.)

2 You ve heard of statistics to deal with problems of uncertainty and differential equations to describe the rates of change of physical systems. In this section, you will learn about two more: vector analysis and matrix algebra. We will discuss both of these subjects somewhat together because they share some of the same terminology and operations. There are many branches of mathematics. In science and engineering, almost all of them are important. (But, they are entirely separate branches of mathematics.) Vector analysis is the mathematics of representing physical variables in terms of coordinate systems. (pic of sky at an angle, with coord system drawn in?) These items could be engineering measurements, prescriptions of a set of algebraic equations, or scores that students received on the last physics test. It defines how information can be represented and operated upon with very efficient notation, but does not necessarily deal with coordinate systems. ruler If we say that the sky is falling, we are implicitly defining a coordinate system. At least we re defining which way is down. grade book alg equation When we speak of velocity or acceleration or position, they are almost always prescribed with respect to some frame of reference, i.e., a coordinate system. Vector analysis gives us the tools to prescribe and manipulate such variables with respect to a frame of reference. It is a part of a broader discipline called tensor analysis. photo of a car, blur? something w/motion blur to lead to the next section (velocity) Matrix algebra is the mathematics of ordered sets or arrays of data and variables. /18/02

3 Velocity and speed are often used interchangeably. But they are entirely different kinds of variables. Speed tells how fas ast, but velocity tells how fast in what direction. Physical variables fall into three classes: scalars,, and tensors. All three are denoted differently and have quite different characteristics. Vector analysis: Attributes of physical variables Think about some physical variables say, mass, velocity, speed, pressure, position. Are they all qualitatively the same? weight/speedometer No. Some of them, like mass and pressure, can be specified with only a single value, a magnitude 20 kg or 150 p.s.i. Others, like velocity and position require you to specify how the variable relates to a frame of reference a coordinate system. These variables are said to have magnitude and direction. Scalars are variables whose value can be expressed purely as a magnitude. Fig. 1 - Scalar, photo of a road. Caption: istance is a scalar. pressure cooker/map Mass, pressure, density, and distance are all scalars; they are represented mathematically by symbols like m, p, ρ, d. Vectors are variables having different values in different directions. Often they re described as having magnitude and direction. There are several different notations for : F, a, u i. The boldface F and underlined a are exactly equivalent. Each represents a triplet of values, one for the projection of the vector onto each of the coordinate directions. The subscripted u i also represents a vector, but is slightly more specific. It suggests that there are three coordinate directions, 1, 2, 3 (corresponding to x, y, z). u i denotes the component of the vector u ( or u) in the i th direction. Or we can be more specific and write u 2 as the component of the u vector in the 2, or y direction. diagram of a vector u (3-?) in i direction When we refer to the variables that are vector quantities, we simple say force vector, displacement vector, or velocity vector. /18/02

4 Tensors are variables whose values are matrices or collections of and require several subscripts for their specification. An example is ε ij the strain tensor which describes the deformation of a material. C ijkl requires 3x3x3x3 values for its prescription. It describes the relationship between the stress tensor σ ij and the strain tensor ε ij. photo of something melted? like a plastic melt? or a microwaved chip bag? : OR... a microwaved Peep ^_^ (with caption) Fig. x Because materials are composed of an organized collection of atoms, a deformation in one direction, say the x- direction, may be accompanied by deformations in both the y- and z-directions as well. And a deformation in the y-direction could produce yet different deformations in the x- and z- directions. To characterize this behavior we need three values in each of three directions. So we describe it in terms of the tensor ε ij. This tensor requires 3 x 3 values to define it, one for each of the possible combinations of i and j. Tensors can get arbitrarily complicated. One other example is the deformation tensor C ijkl. diagram of either Cijkl or the Eij tensor! Eij maybe less complicated. A tensor with two subscripts is called a second-order tensor. (Guess what a tensor with four subscripts is called!) In fact, one can view all variables as tensors. A vector with one subscript is a first-order tensor, and a scalar with no subscripts is a zeroth-order tensor. another diagram, maybe of Cijkl, or, three squares with arrows denoting zeroth-, first-, and second-order tensors We will not use tensors at all in this course. And since they can get complicated very quickly, we will not discuss them further. We introduce them only for completeness. /18/02

5 There are four properties of a vector. It has magnitude and direction, acts along the line of its direction, and has no fixed origin. Vectors Vectors are extremely important in science. Not only are they necessary for describing physical variables, they are also necessary for describing coordinate systems--the framework within which physical things are described. So, what is a vector? What are its properties? photo of something in motion (a bike messenger?), or tour de france (mph is magnitude) There are four. It has magnitude. It has direction. It acts along the line of its direction. It has no fixed origin. The best way to visualize this is with an arrow. above photo w/arrows, or just arrow diagrammed The length of the arrow represents the magnitude of the vector; the orientation of the arrow with respect to a coordinate system represents the direction of the vector. if we can find an arrow-shaped suitcase, or manipulate photo to make it seem like a businessman is holding an arrow suitcase. or, a statue with a spear/javelin Imagine you are in a room (a coordinate system) and you re physically holding an arrow in some position. You can walk around the room holding the arrow in the same orientation with respect to the room, and it would still be the same vector. diagram above photo w/coordinate system, vector Finally, what if the arrow represented, say, velocity? The velocity would be in the direction of the arrow. Although we can talk about without referring to a coordinate system, we almost always define and use within a coordinate system. So let s discuss coordinate systems a little bit. /18/02

Orthogonal: to be at right angles with respect to each other You are all familiar with the Cartesian coordinate system x,y,z, right? Each of the axes is orthogonal to one another, which means that the axes are at 90 angles with respect to one another. And any point in space can be defined in terms of its distance along each of the three axes. That s the idea of a coordinate system: to provide a frame of reference from which to measure and define things. Let s begin with a 2-dimensional example because it s easier to sketch. We would then have simply an x,y coordinate. Suppose we begin with a vector V given in terms of its magnitude V and direction θ. This vector can be represented in terms of its x,y components by geometrically projecting the length of the vector onto each of the two axes. Here s a picture: Since each of the axes of a coordinate system specifies a direction, each axis could itself be represented as a vector. It turns out that any vector U can be completely represented as (or decomposed into) a sum of in any three independent directions. re?diagram of vector U showing three directions This means that a vector U can be expressed directly in terms of representing the coordinate directions. In a Cartesian coordinate system these are denoted by the unit i, j, k. (A unit vector has magnitude one.) Then U might be represented as ai + bj + ck. a,b,c are the magnitudes of the U vector in the i,j,k directions. y θ V projection of V in x-direction x In cartesian coordinates: V x = V cos θ V y = V sin θ The magnitude of U is U =. In the previous example V can be now described in terms of component as V = V x i + V y j. The magnitudes V x and V y scale the i, j unit to represent V. So the components or projections of V onto the x and y axes are the magnitudes V x and V y. But this is a little awkward to deal with mathematically. There s a more convenient way. photo, maybe of a plane taking off, or a golf ball after being hit, diagrammed to relate to vector V. /18/02

7 What can we do with? For one, we can add them and subtract them. Since prescribe magnitude and direction, their addition and subtraction is geometric. That is, to add two A and B move one of the, say B, (but keep its orientation) so that its tail is at the head of A. The line between the tail of A and the head of B will be a new vector C = A + B. It s similar for subtraction, except to get A = C B one moves B with respect to C so that their heads (or tails) come together. Graphically, here s what it looks like. A B C=A+B A=C-B Notice that both of these figures look exactly the same. That s because the vector operations of addition and subtraction behave just like algebra. C = A + B is exactly the same equation as A = C B. B C Notice we said before that for subtracting B from C, either their heads or their tails should be brought together. Sketch this to convince yourself that you get the same answer. These diagrams are presented in only two dimensions, but the same rules apply in three dimensions. sketch here area? print out this page and sketch... not sure of photo. There s another way to add two A and B together, and that s to consider their individual components. You simply add the projections of each of the components together. So, if A = a i + a j + a k 1 2 3 and B = b i + b j + b k, 1 2 3 then C = A + B = (a + b ) i + (a + b ) j + (a + b ) k 1 1 2 2 3 3 /18/02

8 Vectors can always be broken down into three component usually in the directions of the axes of the coordinate system. You should look carefully at both the graphical method and the component method to understand why they re both the same. In practice, sometimes it s practical to use one method, other times the other method. A Let s look at a few more examples. E B C F = A+B+C++E =? F = A+B+C++E =? A C E B Vector multiplication and division are not defined. However, there are two other vector operations that are: vector dot product A B, and vector cross-product A B. Since neither of these concepts will be used in this course, we will not discuss them further. Just be aware that they exist. If you re curious, their definitions can be found in any elementary book on. In Example 1, what s F? Remember, the resulting vector is the oriented line from the tail of the first vector to the head of the last vector. What about Example 2? This one, you should be able to figure out in parts. That s enough of vector analysis. The important thing to remember is can always be decomposed into three component usually in the directions of the axes of the coordinate system. Now let s turn to that other branch of mathematics that deals with : matrix algebra. The answers: F = 0, because the distance between the tail of A and the head of E is zero; and F = E. A and B are the negatives of one another, so they contribute zero. The same is true for C and. That leaves E. 4 squares - relating to matrix algebra in pictures, puter screen from The Matrix, chain-link fence, 3- graphic, video game still (Quake/Tomb Raider), Ghost in the Shell? /18/02

9 In matrix algebra, a vector is an ordered list of elements. There are two types of in matrix algebra: column and row. Matrix algebra Science and engineering often entail lists of data, sets of equations, and coordinate system transformations. If one had to write each individual element in a derivation or a presentation, discourses would be tediously long. And the essential ideas could be swamped by details. To present these ideas more practically, we can use the more compact notation of and matrices. And with this compact notation we can perform mathematical operations using matrix algebra. First, what is a vector in matrix algebra? This will be a slightly different definition than before. new defn of vector diagram A vector is an ordered list of elements. It s that simple. There are no references to coordinates systems. The list could be 10 pieces of recorded data, the 15 unknowns in a set of 15 simultaneous equations, or the 4 coefficients of an equation. What is important is that the list is in some order, and the elements of the list can be identified by their location within the list. In each of these cases we can represent the information by a vector name, say a, and an index. Then the vector, or array of information might be a = (1, 3, 5, 2). So the third piece of recorded data could be identified as a 3 (Here, a 3 = 5). And any piece of data could be referred to as a i. No one will misunderstand us if we identify an array or vector as a i. But, in matrix algebra, if we want to perform mathematical operations, we must be more specific and distinguish between two types of : column, and row. Each of these could contain our list of data, so it would seem unnecessary to require two types to hold the same information. But, later we will define some rules of matrix algebra that will require us to distinguish between row and column. Using the example of the array a from before, a would be represented as a column vector 1 3 a=. 5 2 As a row vector, A would be represented as a T =(1 3 5 2). Notice the transpose sign the T. This means that the rows in vector a become the columns in vector a T. So we can change the representation of an array a from a column vector a to a row vector a T just be adding the superscript T. /18/02

10 A two-dimensional array is called a matrix. a matrix can refer to almost any information, whereas a tensor only refers to information relative to a coordinate system. Now, a matrix. Sometimes we will need to represent a two-dimensional array of information, for example, the 4 coefficients of each equation for 4 simultaneous equations. A two-dimensional array is called a matrix. Just like the vector, we give it a name, say C. For matrices, we ll use uppercase, bold letters. However, sometimes you will also see the notation C. (Just like a vector might have a single underline, a matrix could be identified with a double underline.) And like a tensor, since there is a matrix of values rather than a linear array, an individual entry will be identified using two subscripts the first to designate which row; the second to designate which column. So, C 3,4 would be the element in row 3 column 4. An example of C might be 1 5 8 2 3 4 4 1 4 2 8 4 9. 3 2 Here, C 3,4 = 3. A matrix need not be square, i.e., the same number of rows as columns. It can be 4 x 2, or 200 x 3, or even 100 x 1 but then it would be a matrix representation of a column vector. Sometimes a column vector is called a column matrix just to remind us that a vector is just a special case of a matrix. illustration of diff btw matrix and tensor. Note that a matrix is similar to a 2 nd order tensor. The difference is that a matrix can have any number of elements, whereas a tensor can only be 3x3. And a matrix can refer to almost any information, whereas a tensor refers only to information relative to a coordinate system. We identify the size of the matrix with the phrase: a matrix of order 3x4. Matrices can also be transposed. This means that the rows are interchanged with columns and vice-versa. It the same operation that can be carried out with. For example, if 2 1 9 A =, 3 4 then 2 3 A T = 1 4. 9 /18/02

11 Certain matrices are given special names. Several of the more important ones are: 1) square matrix the number of rows equals the number of columns, e.g., A = 2, 3 rows, 3 columns 2) symmetric matrix a matrix in which the elements A i,j = A j,i, e.g., A = 3) diagonal matrix a matrix in which the only non-zero elements are along the main diagonal, A i,j = 0, for i j. The main diagonal are the elements A i,i. A = 2 5 7 2 1 4 2 0 0 1 2 3 0 1 0 1 0 3 0 0 4 4 3 3 4) the unit or identity matrix a diagonal matrix with 1 s down the main diagonal. 1 0 0 A = 0 1 0 = I 0 0 1 The letter I is always used to refer to the identity matrix I. pull definition of identity matrix, or definitions of all 4 important types - square, symmetric, diagonal, identity The one other concept we need for matrix algebra is the scalar identical to that in vector analysis. It is just a simple number or magnitude, like 3. In fact, all these representations of information twodimensional arrays, one-dimensional arrays, and simple numbers can be considered as different levels of matrices. A two-dimensional array is a second-order matrix, i.e., it requires two subscripts to identify a particular value; a onedimensional array is a first-order matrix; and a scalar is a zeroth-order matrix. Sound familiar? /18/02

12 Unfortunately, the word order is used in many different ways in mathematics. An ordered set of elements refers to the organization of information in an array. A 2 nd order matrix is one that requires two subscripts for specification. A matrix of order 5x3 is a matrix with 5 rows and 3 columns. A 3 rd order polynomial has four terms (sic). You get used to it... Now, we need a set of rules for how we can carry out mathematics with scalars, and matrices. Equality: 1. a = b if and only if a and b are the same length and the corresponding elements are identical, i.e., a i = b i. The same rules apply if a T = c T. 2. = F if and only if and F are the same size and the corresponding elements are identical, i.e., i,j = F i,j. 1 4 1 4 If A =, and if B =, then A = B. 3 2 3 2 Matrix operations: 1. Addition, subtraction. These operations are carried out on corresponding members of like /matrices. So, c = a + b c i = a i + b i c T = a T - b T c i = a i - b i G = + F G i,j = i,j + F i,j 2. Multiplication. To obtain the product G = F, the number of columns in (the first matrix) must equal the number of rows in F (the second matrix). If is of order M x N and F is of order N x P, then G will be of order M x P. F = G (M x N) (N x P) (M x P) Note: multiplication does not commute. That is, F will not, in general, give the same result as F. In fact, F may not even be a permissible operation. /18/02

13 Multiplication is defined by the operation G = i,j N k = 1 i F, k k, j We can also represent this operation as follows for G 2,2 : A row vector can be construed as a matrix of order 1 x N; a column vector can be construed as a matrix of order M x 1. So matrix/vector multiplication is defined, provided the number of columns in the first matrix/vector is equal to the number of rows in the second matrix/vector. 1,1 2,1 3,1 1,2 2,2 3,2 1,3 2,3 3,3 F F F 1,1 2,1 3,1 F F F 1,2 2,2 3,2 Some examples: Let a be a 1 x 3 column vector; let b be a 1 x 4 column vector; and let be a 3 x 4 matrix. Then the following multiplication operations are valid: G 2,2 = 2,1 F 1,2 + 2,2 F 2,2 + 2,3 F 3,2 To obtain all the elements of G, we must carry out this process for each combination of rows in times columns in F. In this case, the result G will be of order 3 x 2. Note that in this example F is not a defined calculation because the number of columns in F does not equal the number of rows in. a c, a 1 x 4 column vector b T e T, a 3 x 1 row vector a a T E, a 3 x 3 matrix b T b s, a 1 x 1 matrix, i.e., a scalar Multiplication of a scalar times a matrix/vector is also possible. The multiplication operation is valid for as well. If s is a scalar, then s = s i,j for all i and j, i.e., every element of the matrix is multiplied by s. The same rule applies to. /18/02

14 3. Inverse. If a matrix is square, then it may have an inverse -1. will have an inverse if the determinant of, 0. The inverse is defined as follows: Suppose there is a square matrix F such that F = I, the identity matrix. The determinant of is represented as and is a single value. How the determinant is actually calculated will not be presented. We introduce it here only because we want to refer to the inverse matrix, say -1. And -1 does not exist if = 0. Then F is said to be the inverse of or -1. were simple variables, then If matrices In the next section, we will relate the determinant to systems of linear equations. 1 = = 1 = I or 1. But, since matrix division does not exist, we write -1 to represent the inverse or reciprocal of. 4. eterminant. Its definition is beyond the scope of this discussion, but we can offer an approximate meaning. First, determinants apply only to square matrices. If the rows of a matrix are thought to be a collection of row or column, then the determinant is an indicator of how similar those are the more similar they are, the closer to zero the determinant will be. If two rows are identical, or if one is a just a constant multiple of another, then the determinant is zero. 5. Matrix operations on equations. Suppose we have the following matrix equation Ax = b, where A is 3 x 3, x is 3 x 1, and b is 3 x 1. That is, each side reduces to a 3 x 1 matrix (or vector). To keep this equation valid, any operation we carry out on the left-hand side of the equation must also be carried out on the right-hand side of the equation no different than in regular algebra. However, multiplication is not commutative in matrix algebra. So we can t say multiply both sides of the equation by the 3 x 3 matrix. We must be more specific. We must say pre-multiply both sides of the equation... or post-multiply both sides of the equation... These would lead, respectively, to Ax = b and Ax = b. /18/02

15 This is pretty heavy stuff. And we ve given almost no examples. How is all of this matrix notation and its associated mathematics useful to us? In this course, we will use it primarily to represent systems of equations. Consider the following set of simultaneous linear algebraic equations, where X,Y,Z have nothing to do with coordinate systems: 3X + 5Y + 7Z = 15 2X - 4Y - 3Z = 9 X + 2Y + Z = 1 Each side of Ax = b is a 3 x 1 column vector. Since the two sides are equal to one another, i.e., the two column are equal, then each corresponding element of the two must be equal. So, that says that the left hand side of row one must equal 3X + 5Y + 7Z, and the right-hand side of row one must equal 15. But that is the statement that 3X + 5Y + 7Z = 15 the first of the three equations. Rows two and three produce the remaining two equations. So, now we have a way to represent a system of equations. How do we solve them? Now, consider the following matrix and : 3 5 7 X 15 A = 2 4 3, x = Y, b = 9. 1 2 1 Z 1 Then the matrix equation Ax = b is exactly equivalent to the three simultaneous equations above. A is the coefficient matrix, x is the vector of unknowns, and b is the right-hand-side vector. (With all of the specialized vocabulary in mathematics, one would think that a better word could be used to describe the right sides of equations.) Using the matrix representation, we can presume that A has an inverse A -1. This will be true if A 0. A will be nonzero, if the three equations are independent, i.e., if none of the equations are linear combinations of the others. Now, let s pre-multiply each side of our equation by A -1. Then, Ax = b A -1 Ax = A -1 b. But since A -1 A = I, we have Ix = A -1 b x = A -1 b. We have solved for x, the vector of unknowns. And its solution is the product of the inverse of the coefficient matrix A times the right-hand side vector b. /18/02

1 There is much more to matrix algebra than what we have presented here. But, even by knowing these few elements, we ll be able to use matrix algebra as a convenient tool for representing and solving many engineering problems. /18/02