# Linear Algebra. Preliminary Lecture Notes

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1 Linear Algebra Preliminary Lecture Notes Adolfo J. Rumbos c Draft date April 29, 23

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3 Contents Motivation for the course 5 2 Euclidean n dimensional Space 7 2. Definition of n Dimensional Euclidean Space Algebraic Structure on Euclidean Space Linear Combinations and Spans Linear Independence Subspaces of Euclidean Space Finitely Generated Subspaces Connections with Systems of Linear Equations Elementary Row Operations Gaussian Elimination Fundamental Theorem of Homogenous Systems Nonhomogeneous Systems Maximal Linearly Independent Subsets Bases Dimension Coordinates Euclidean Inner Product and Norm Definition of Euclidean Inner Product Euclidean Norm Orthogonality Spaces of Matrices Vector Space Structure in M(m, n) Matrix Algebra The row column product The product of a matrix and a vector Interpretations of the matrix product Ax The Matrix Product Properties of Matrix Multiplication Invertibility Right and Left Inverses Definition of Inverse

4 4 CONTENTS Constructing Inverses Nullity and Rank Linear Transformations 9 4. Vector Valued Functions Linear Functions Matrix Representation Compositions Orthogonal Transformations Areas, Volumes and Orientation Areas of Parallelograms Determinant of 2 2 matrices Orientation in R The Cross Product The Triple Scalar Product Determinant of 3 3 matrices Orientation in R The Eigenvalue Problem Reflections in R Rotations in R

5 Chapter Motivation for the course Imagine a ball whose center is at a fixed location in three dimensional space, but is free to rotate about its center around any axis through the center. The center of the ball is not allowed to move away from its fixed location. Imagine that we perform several rotations about various axes, one after the other. We claim that there are two antipodal points on the surface of the ball which are exactly at the same locations they were at the beginning of the process. Furthermore, the combination of all the rotations that we perform has the same affect on the ball as that of a single rotation performed about the axis going through the fixed antipodal points. This result is know in the literature as Euler s Theorem on the Axis of Rotation (see [PPR9]). One of the goals of this course will be the proof if this fact. We will require all of the machinery of Linear Algebra to prove this result. The machinery of Linear Algebra consists of a new language we need to learn, new concepts we need to master and several theorems that we need to understand. The language and concepts of Linear Algebra will help us find convenient ways to represent rotations in space. Rotations, we will see, are special kinds of linear 5

6 6 CHAPTER. MOTIVATION FOR THE COURSE transformations, which are functions that map points in space to points in space and which satisfy some special properties. We have studied functions in Calculus already. In Calculus I and II we dealt with real valued functions defined on intervals of the real line, R; in Calculus III, we learned about functions which may be defined in regions of the plane, R 2, or three dimensional space, R 3, and which may be real valued or vector valued (also known as vector fields). In Linear Algebra we focus on a class of functions which are defined in all of the space (one, two, or three dimensional space, or higher dimensional space) and can take on values in a one dimensional or higher dimensional space. The functions we will deal with have the property known as linearity. Loosely speaking, linearity means that the functions interact nicely with the algebraic structure that the spaces on which the functions act have: the structure of a linear space or a vector space. The study of vector spaces will be one of the major topics of this course. We begin our discussion of vector spaces by introducing the example of Euclidean n dimensional space. The main concepts of Linear Algebra will first be defined in the context of Euclidean space and then will be presented in more general context later on in the course.

7 Chapter 2 Euclidean n dimensional Space 2. Definition of n Dimensional Euclidean Space Euclidean space of dimension n, denoted by R n in this course, will consist of columns of real numbers of the form x x 2.. x n These are called column vectors. In many textbooks elements of R n are denoted by row vectors; in the lectures and homework assignments, we will use column vectors to represents the elements in R n. Vectors in R n can be used to locate points in n dimensional space. They can also be used to indicate displacements in a certain direction and through certain distance. Example 2... Consider two dimensional space, R 2. This can be represented by the familiar xy plane ( ) pictured ( ) in Figure The vectors and are represented in the figure as arrows, or 2 directed line segments, emanating from the origin of the xy plane. ( ) In the previous example, the vector can be used to locate a point in the 2 xy plane with coordinates (, 2). However, it can also indicate a displacement from the origin to the point (, 2) through the straight line segment joining them. Notation (Vector Notation and Conventions). In the lectures and in these notes we will use the symbols u, v, w, etc. to denote vectors. In several linear algebra 7

8 8 CHAPTER 2. EUCLIDEAN N DIMENSIONAL SPACE y ( ) 2 ( ) x 2 Figure 2..: Two dimensional Euclidean Space texts, though, these symbols are usually written in boldface, u, v, w, etc., or with an arrow on top of the letter, u, v, w, etc. In these notes, real numbers will usually be denoted by the letters a, b, c, d, t, s, x, y, x, etc. and will be called scalars in order to distinguish them from vectors. I will also try to follow my own convention that if we are interested in locating a point in space, we will use the row vector made up of the Cartesian coordinates of the point; for instance, a point P in R n will be indicated by P (x, x 2,..., x n ), where x, x 2,..., x n are the coordinates of the point. As mentioned earlier, vectors in R n can also be used to indicate displacement along a straight line segment. For instance, the point P (x, x 2,..., x n ) is located by the vector v = OP = where O denotes the origin, or zero vector, in n dimensional Euclidean space. The arrow over the symbols OP emphasizes the displacement nature of the vector v. ( ( ) 2 Example Denote the vectors and in Figure 2.. by v 2) and v 2, respectively. Then, v and v 2 locate the point P (, 2) and P 2 (2, ), respectively. See Figure Note, however, that the arrow representing the vector v 2 in Figure 2..2 does not have to be drawn with its starting point at the origin. It can be drawn anywhere as long as its length and direction are the same (see Figure 2..2). We will still call it the vector v 2. Only when the base of the arrow representing v 2 is located at the origin will it be locating the point P 2 (2, ). In all other instances, the vector v 2 represents a displacement parallel to that from the origin to the point (2, ). x x 2. x n,

9 2.2. ALGEBRAIC STRUCTURE ON EUCLIDEAN SPACE 9 y (, 2) v v 2 2 v v 2 v 2 x (2, ) Figure 2..2: Dual Nature of Vectors in Euclidean Space 2.2 Algebraic Structure on Euclidean Space What makes R n into a vector space are the algebraic operations that we will define in this section. We begin with vector addition.. Vector Addition x y x 2 Given v =. and w = y 2, the vector sum v + w or v and w is. x n y n x + y x 2 + y 2 v + w =. x n + y n ( ( ) 2 Example Let v = and v 2) 2 =. Then, the vector sum of v and v 2 is ( ) ( ) v + v 2 = =. 2 Figure shows a geometric interpretation of the vector sum of the vectors v and v 2 in the previous example. It is known as the parallelogram rule: the arrow representing the vector v 2 is drawn with its base at the tip of the arrow representing the vector v. The vector sum v +v 2 is then represented by the arrow going from the base of the arrow representing v to the tip of the translated arrow representing v 2. Notice that we could have obtained the same vector sum, v + v 2, if, instead of translating the arrow representing v 2, we would have translated

10 CHAPTER 2. EUCLIDEAN N DIMENSIONAL SPACE y v 2 v v + v 2 v 2 x Figure 2.2.3: Parallelogram Rule the arrow representing v to the tip of the arrow representing v 2 ; see Figure y v 2 v v + v 2 v 2 v x Figure 2.2.4: Commutative Property for Vector Addition The picture in Figure illustrates the fact that v + v 2 = v 2 + v. This is known as the commutative property of vector addition, which can be derived algebraically from the definition and the fact that addition x 2 of real numbers is commutative: for any vectors v =. and w = x n x

11 2.2. ALGEBRAIC STRUCTURE ON EUCLIDEAN SPACE y y 2. in Rn, y n y + x x + y y 2 + x 2 w + v =. = x 2 + y 2. = v + w. y n + x n x n + y n Properties of Vector Addition Let u, v, w denote vectors in R n. Then, (a) Commutativity of Vector Addition v + w = w + v (b) Associativity of Vector Addition (u + v) + w = u + (v + w) Like commutativity, this property follows from the definition and the fact that addition of real numbers is associative: x y x 2 Write v =., w = y 2. and u = z 2.. Then, x n y n z n z + x y (z + x ) + y z 2 + x 2 (u + w) + v =. + y 2. = (z 2 + x 2 ) + y 2.. z n + x n y n (z n + x n ) + y n Thus, since (z i + x i ) + y i = x i + (x i + y i ), for each i =, 2,..., n, by associativity of addition of real numbers, it follows that (z + x ) + y z + (x + y ) (z 2 + x 2 ) + y 2 (u+w)+v =. = z 2 + (x 2 + y 2 ) = u+(v +w).. (z n + x n ) + y n z n + (x n + y n ) (c) Existence of an Additive Identity The vector =.. in Rn has the property that z v + = + v = v for all v in R n.

12 2 CHAPTER 2. EUCLIDEAN N DIMENSIONAL SPACE This follows from the fact that x + = x for all real numbers x. (d) Existence of an Additive Inverse x x x 2 Given v =. in x 2 Rn, the vector w defined by v =. has x n x n the property that v + w =. The vector w is called an additive inverse of v. 2. Scalar Multiplication x 2 Given a real number t, also called a scalar, and a vector v =., the x n scaling of v by t, denoted by tv, is given by tx tx 2 tv =. tx n ( Example Given the vector v = in R 2) 2, the scalar products ( ) v and v are given by ( ) v = 2 ( ) /2 and 3 2 v = ( ) 3/2, 3 respectively. The arrows representing these vectors are shown in Figure Observe that the arrows representing the scalar products of v lie on the same line as the arrow representing v. Properties of Scalar Multiplication (a) Associativity of Scalar Multiplication Given scalars t and s and a vector v in R n, t(sv) = (ts)v. This property follows from the definition of scalar multiplication and the fact that s(tx) = (st)x for all real numbers x; that is, associativity of multiplication of real numbers. x

13 2.3. LINEAR COMBINATIONS AND SPANS 3 y v 2 v 3 2 v x Figure 2.2.5: Scalar Multiplication (b) Identity in Scalar Multiplication The scalar has the property that 3. Distributive Properties v = v for all v R n. Given vectors v and w in R n, and scalars t and s, (a) t(v + w) = tv + tw (b) (t + s)v = tv + sv. These properties follow from the distributive properties for addition and multiplication in the set of real numbers; namely t(x + y) = tx + ty for all t, x, y R, and respectively. (t + s)x = tx + sx for all t, s, x R, 2.3 Linear Combinations and Spans Given a vector v in R n, the set of all scalar multiples of v is called the span of the set {v}. We denote the span of {v} by span({v}). In symbols, we write span({v}) = {tv t R}. Geometrically, if v is not the zero vector in R n, span{v} is the line through the origin on R n in the direction of the vector v.

14 4 CHAPTER 2. EUCLIDEAN N DIMENSIONAL SPACE Example 2.3. (In R 3 ). Let v = 2. Then, span{v} = t 2 t R We can also write this set as span{v} = x y R n x y = t 2t, t R z z t Figure shows a sketch of the line in R 3 representing span{v}. z x v y span{v} Figure 2.3.6: Line in R 3 x Note that y is a vector on the line, span{v}, if and only if z x t y = 2t z t for some scalar t. In other words, x y is on the line if and only if the z coordinates x, y and z satisfy the equations x = t y = 2t z = t.

15 2.3. LINEAR COMBINATIONS AND SPANS 5 These are known as the parametric equations of the line and t is called a parameter. Definition (Linear Combinations). Given vectors v, v 2,..., v k in R n, the expression c v + c 2 v c k v k, where c, c 2,..., c k are scalars, is called a linear combination of the vectors v, v 2,..., v k. Definition (Span). Given vectors v, v 2,..., v k in R n, the collection of all linear combinations of the vectors v, v 2,..., v k is called the span of the set of vectors {v, v 2,..., v k }. We denote the span of {v, v 2,..., v k } by We then have that span{v, v 2,..., v k }. span{v, v 2,..., v k } = {t v + t 2 v t k v k t, t 2,..., t k R}. Example Consider the vectors v and v 2 in R 3 given by v = and v 2 =. 2 Let s compute span{v, v 2 }. Solution: Write span{v, v 2 } = {c v + c 2 v 2 c, c 2 R} = = c + c 2 c, c 2 R 2 c c 2 c + c, c 2 R c 2c 2 c + c 2 = c c, c 2 R. c + 2c 2 x We then have that a vector y is in span{v, v 2 } if and only if z x y = c + c 2 c z c + 2c 2

16 6 CHAPTER 2. EUCLIDEAN N DIMENSIONAL SPACE for some scalars c and c 2 ; that is, if c + c 2 = x c = y c + 2c 2 = z. Substituting the second equation, c = y, into the first and third equation leads to the two equation { y + c2 = x y + 2c 2 = z. Solving for c 2 in the first equation and substituting into the second yields the single equation 2x y z =. This is the equation of a plane through the origin in R 3 and containing the points with coordinates (,, ) and (,, 2). x In the previous example we showed that if a vector y is in the span, z W = span,, 2 of the vectors v = and v 2 = in R 3, then it determines a point 2 with coordinates (x, y, z) in R 3 lying in the plane with equation 2x y z =. Denote the plane by Q; that is, x Q = y R 3 2x y z =. z Then, the previous example shows that W is a subset of Q. We write W Q, meaning that every element in W is also an element in Q. We will presently show that Q is also a subset of W ; that is, every point in the plane Q must also be in the span of the vectors v = and v 2 =. 2 Example Let W = span, 2

17 2.3. LINEAR COMBINATIONS AND SPANS 7 and Show that Q W. Q = x y R 3 2x y z =. z Solution: To show that Q is a subset of W, we need to show that everypoint in the plane Q is a linear combination of the vectors v = and v 2 =. 2 x Thus, let y Q. Then, z 2x y z =. Solving for z in terms of x and y in the previous equation yields z = 2x y. Thus, z depends on both x and y, which can be thought of as parameters. We therefore set x = t and y = s, where t and s are parameters. We then have x that, if y Q, then z x = t y = s z = 2t s. x In vector notation, we then have that, if y Q, then z x y = z = t s 2t s t + s, 2t s where we have used the definition of vector addition in R 3. Thus, using now the definition of scalar multiplication, we get that x y = t + s z 2,

18 8 CHAPTER 2. EUCLIDEAN N DIMENSIONAL SPACE which shows that, if x y Q, then z x y span, = span v 2,. z 2 x In order to complete the proof that y span{v, v 2 }, we will need to show z that the vector is in the span of the vectors v and v 2 ; that is, we need to find scalars c and c 2 such that c v + c 2 v 2 =, or c + c 2 c =. c + 2c 2 This leads to the system of equations c + c 2 = c = c + 2c 2 =, which has solution: c =, c 2 =. Thus, = v v 2. x Consequently, if y Q, then z x y = c v 2 + c 2 z for some scalars c and c 2, by what we have seen in the first part of this proof. Hence, since = v v 2, it follows that x y = c v 2 + c 2 (v v 2 ) = c 2 v + (c 2 c )v 2, z

19 2.4. LINEAR INDEPENDENCE 9 which is a linear combination of v and v 2. Hence, x x y Q y span{v, v 2 } = W. z z We have therefore shown that Q W. The previous two examples show that the span of v and v 2 is the same set as the plane in R 3 with equation 2x y z =. In other words, the combination of the statements W Q and Q W is equalivalent to the statement W = Q. 2.4 Linear Independence In the previous example we showed that the vector v 3 = of the set {v, v 2 }, where v = and v 2 =. 2 is in the span When this happens (i.e., when one vector in the set is in the span of the other vectors) we say that the set {v, v 2, v 3 } is linearly dependent. In general, we have the following definition: Definition 2.4. (Linear Dependence in R n ). A set of vectors, S, in R n is said to be linearly dependent if at least one of the vectors in S is a finite linear combination of other vectors in S. Example We have already seen that the set S = {v, v 2, v 3 }, where v =, v 2 =, and v 3 =, 2 is a linearly dependent subset of R 3 since v 3 = v v 2 ; that is, v 3 is in the span of the other vectors in S.

20 2 CHAPTER 2. EUCLIDEAN N DIMENSIONAL SPACE Example Let v, v 2,..., v k be any vectors in R n. Then, the set S = {, v, v 2,..., v k }, where denotes the zero vector in R n, is linearly dependent since = v + v v k ; that is, is in the span of the vectors v, v 2,..., v k. If a subset, S, of R n is not linear dependent, we say that it is linearly independent. Definition (Linear Independence in R n ). A set of vectors, S, in R n is said to be linearly independent if it is not linearly dependent; that is, no vector in S can be expressed as a linear combination of other vectors in S. The following proposition gives an alternate characterization of linear independence for a finite subset of R n. Proposition The set S = {v, v 2,..., v k } of vectors in R n is linearly independent if and only if c =, c 2 =,..., c k = is the only solution to the vector equation c v + c 2 v c k v k =. Remark Note that it is not hard to see that c =, c 2 =,..., c k = is a solution to the equation c v + c 2 v c k v k =. (2.) The solution c =, c 2 =,..., c k = is usually referred to as the trivial solution. Thus, linear independence is equivalent to the statement that the trivial solution is the only solution to the equation in (2.). Thus, linear dependence of the set {v, v 2,..., v k } is equivalent to the statement that the equation c v + c 2 v c k v k = has solutions in addition to the trivial solution. Remark The statement of Proposition is a bi conditional; that is, it is the combination of the two implications:. If the set S = {v, v 2,..., v k } is linearly independent, then c =, c 2 =,..., c k = is the only solution to the vector equation c v + c 2 v c k v k = ;

21 2.4. LINEAR INDEPENDENCE 2 2. Conversely, if c =, c 2 =,..., c k = is the only solution to the vector equation c v + c 2 v c k v k =, then S = {v, v 2,..., v k } is linearly independent. Thus, in order to prove Proposition 2.4.5, the two implications need to be established. We will now prove Proposition This is the first formal proof that we present in the course and will therefore be presented with lots of details in order to illustrate how a mathematical argument is presented. Subsequent arguments in these notes will not be as detailed as this one. Proof of Proposition We first prove that if the set S = {v, v 2,..., v k } is linearly independent, then c =, c 2 =,..., c k = is the only solution to the vector equation c v + c 2 v c k v k =. Suppose therefore that S is linearly independent. This means that no vector in S is in the span of the other vectors in S. We wish to prove that the equation has only the trivial solution c v + c 2 v c k v k = c =, c 2 =,..., c k =. If this is not the case, then there exist scalars c, c 2,..., c k, such not all of them are zero and c v + c 2 v c k v k =. Suppose the non zero scalar is c j, for some j in {, 2,..., k}, and write c v + c 2 v c j v j + c j v j + c j+ v j+ + + c k v k =. (2.2) We can solve for c j v j in equation (2.2) by adding on both sides the additive inverses of the other vectors. Using the properties of vector addition we then get that c j v j = c v c 2 v 2 c j v j c j+ v j+ c k v k,

22 22 CHAPTER 2. EUCLIDEAN N DIMENSIONAL SPACE which, using now the properties of scalar multiplication can now be re written as c j v j = ( c )v +( c 2 )v 2 + +( c j )v j +( c j+ )v j+ + +( c k )v k. (2.3) Now, since c j, /c j exists. We can then multiply both sides of equation (2.3) by /c j, and using now the distributive properties and the associative property for addition and scalar multiplication we obtain that ( ) c v j = v + + c j ( cj c j ) v j + ( c j+ c j ) v j+ + + ( ck c j ) v k. (2.4) Equation (2.4) displays v j as a linear combination of v,..., v j, v j+,..., v k. However, this is impossible since we are assuming that S is linearly independent and therefore no no vector in S is in the span of the other vectors in S. This contradiction then implies that the equation has only the trivial solution c v + c 2 v c k v k = c =, c 2 =,..., c k =, which we had set out to prove. Next, we prove the converse statement: if c =, c 2 =,..., c k = is the only solution to the vector equation c v + c 2 v c k v k =, (2.5) then S = {v, v 2,..., v k } is linearly independent. Suppose that c v + c 2 v c k v k = has only the trivial solution c =, c 2 =,..., c k =. Arguing by contradiction again, assume that S is not linearly independent. Then, one of the vectors in S, say v j, is in the span of the other vectors in S; that is, there exist scalars c, c 2,..., c j, c j+,..., c k such that v j = c v + c 2 v c j v j + c j+ v j+ + + c k v k. (2.6) Adding the additive inverse to both sides of equation (2.6) we obtain that c v + c 2 v c j v j v j + c j+ v j+ + + c k v k =,

23 2.4. LINEAR INDEPENDENCE 23 which may be re written as c v + c 2 v c j v j + ( )v j + c j+ v j+ + + c k v k =. (2.7) Since, equation (2.7) shows that there is a non trivial solution to the equation c v + c 2 v c k v k =. This contradicts the assumption that the only solution to the equation in (2.5) is the trivial one. Therefore, it is not the case that S is linearly dependent and hence it must be linearly independent. Proposition is very useful in determining whether a given set of vectors, {v, v 2,..., v k }, in R n is linearly independent or not. According to Proposition 2.4.5, all we have to do is to solve the equation c v + c 2 v c k v k = and determine whether it has one solution or more than one solution. In the first case (only the trivial solution) we can conclude by virtue of Proposition that the set is linearly independent. In the second case (more than one solution), the set is linearly dependent. Example Determine whether the set, independent in R 3 or not., 2 2 is linearly Solution: Consider the equation c + c 2 + c 3 2 =. (2.8) 2 This equation leads to the system of linear equations c + c 2 + c 3 = c + 2c 3 = c + 2c 2 =, (2.9) Solving for c 3 in the first equation and substituting into the second equation leads to the system of two equations { c 2c 2 = (2.) c + 2c 2 =. Observe that the system of equations in (2.) is really a single equation in two unknowns c + 2c 2 =. (2.)

24 24 CHAPTER 2. EUCLIDEAN N DIMENSIONAL SPACE We can solve for c in equation (2.) and c 3 in the first equation n (2.9) to obtain that c = 2c 2 (2.2) c 3 = c 2, which shows that the unknown scalars c and c 3 depend on c 2, which could taken on arbitrarily any value. To stress the arbitrary nature of c 2, let s rename it t, an arbitrary parameter. We then get from (2.2) that c = 2t c 2 = t c 3 = t. (2.3) Since the parameter t in (2.3) is arbitrary, we see that the system in (2.9) has infinitely many solutions. In particular, the vector equation (2.2) has non trivial solutions. Itthen follows by virtue of Proposition that the set,, 2 is linearly dependent. 2 Example Determine whether the set, independent in R 3 or not. Solution: Consider the equation c + c c 3 =, 2 is linearly. (2.4) This equation leads to the system of linear equations c + c 2 = c + 2c 3 = c + 2c 2 + c 3 =, (2.5) Solving for c and c 2 in the first two equations in (2.5) leads to c = c 2 =. Substituting for these in the third equation in (2.5) then leads to c 3 =. We have therefore shown that the vector equation in (2.4) has only the trivial solution. Consequently, by virtue of Proposition that the set,, 2 is linearly independent.

25 2.5. SUBSPACES OF EUCLIDEAN SPACE 25 Remark In the previous two examples we have seen that the question of whether a finite set of vectors in R n is linearly independent or not leads to the question of whether a system of equations, like those in (2.9) and (2.5), has only the trivial solution or not. The systems in (2.9) and (2.5) are examples of homogeneous systems. In general, a homogenous system of linear of m equations in n unknowns is of the form a x + a 2 x a n x n = a 2 x + a 22 x a 2n x n =. =. a m x + a m2 x a mn x n =, (2.6) where the x, x 2,..., x n are the unknowns, and a ij, for i =, 2,..., m and j =, 2,..., n, are known coefficients. We will study systems more systematically later in the course and we will see that what is illustrated in the previous two examples is what happens in general: either the linear homogenous system has only the trivial solution, or it has infinitely many solutions. 2.5 Subspaces of Euclidean Space In this section we study some special subsets of Euclidean space, R n. These are called subspaces and are defined as follows Definition 2.5. (Subspaces of R n ). A non empty subset, W, of Euclidean space, R n, is said to be a subspace of R n iff (i) v, w W implies that v + w W ; and (ii) t R and v W implies that tv W. If (i) and (ii) in Definition 2.5. hold, we say that the set W is closed under the vector space operations in R n. For this reason, properties (i) and (ii) are usually referred to as closure properties. There are many examples of subspaces of R n ; but there are also many examples of subsets of R n which are not subspaces. We shall begin by presenting a few examples of subsets which are not subspaces. Example (Subsets which are not subspaces).. The empty set, denoted by the symbol, is not a subspace of any Euclidean space by definition. 2. Consider the subset, S, of R 2 given by the first quadrant in the xy plane: S = {( ) } x R 2 x, y y

26 26 CHAPTER 2. EUCLIDEAN N DIMENSIONAL SPACE S is not a subspace since ( ) S, but ( ) ( ) = ( ) is not in S because <. That is, S is not closed under scalar multiplication. 3. Let S R 2 this time be given by {( ) } x S = R 2 xy. y In this case, S is closed under scalar multiplication, ( but it is not closed x under vector addition. To see why this is so, let S. Then, xy. y) Then, for any scalar t, note that (tx)(ty) = t 2 xy since t 2 for any real number t. Thus, S is closed under scalar multiplication. However, S is not closed under vector addition; to see this, consider the vectors ( ( ) v = and w =. ) Then, v and w are both in S since However, = ( ) =. v + w = is not in S since ( ) = <. ( ) Example (Subsets which are subspaces).. Let W = {}; that is, W consists solely of the additive identity,, in R n. W is a subspace of R n because + = W, so that W is closed under vector addition; and t = W for all t R; that is, W is closed under scalar multiplication.

27 2.5. SUBSPACES OF EUCLIDEAN SPACE W = R n, the entire Euclidean space, is also a subspace of R n. 3. Let W = x y ax + by + cz =, where a, b and c are real numbers, z is a subspace of R 3. Proof: Let v = x y and w = x 2 y 2 be in W. Then, z z 2 Adding both equations yields ax + by + cz = ax 2 + by 2 + cz 2 =. a(x + x 2 ) + b(y + y 2 ) + c(z + z 2 ) =, where we have used the distributive property for real numbers. It then follows that x + x 2 v + w = y + y 2 W, z + z 2 and so W is closed under vector addition in R 3. Next, multiply ax + by + cz = on both sides by a scalar t and apply the distributive and associative properties for real numbers to get that a(tx ) + b(ty ) + c(tz ) =, which show that tx tv = ty W, tz and therefore W is also closed with respect to scalar multiplication. Hence, W is closed with respect to the vector space operations in R 3 ; that is, W is a subspace of R n. Let S = {v, v 2,..., v k } be a subset of R n and put W = span(s). Then, W is a subspace of R n. Proposition Given a non empty subset, S, of R n, span(s) is a subspace of R n.

28 28 CHAPTER 2. EUCLIDEAN N DIMENSIONAL SPACE Proof: Since S, there is a vector v S. Observe that v = v is a linear combination of a vector from S; therefore, v span(s) and therefore span(s) is non empty. Next, we show that span(s) is closed under the vector space operations of R n. Let v span(s); then, there exist vectors v, v 2,..., v k in S such that v = c v + c 2 v c k v k for some scalars c, c 2,..., c k. Thus, for any scalar t, tv = t(c v + c 2 v c k v k ) = t(c v ) + t(c 2 v 2 ) + + t(c k v k ) = (tc )v + (tc 2 )v (tc k )v k, which shows that tv is a linear combination of elements in S; that is, tv span(s). Consequently, span(s) is closed under scalar multiplication. To show that span(s) is closed under vector addition, let v and w be in span(s). Then, there exist vectors v, v 2,..., v k and w, w 2,..., w m in S such that v = c v + c 2 v c k v k and w = d w + d 2 w d m w m, for for some scalars c, c 2,..., c k and d, d 2,..., d m. Thus, v + w = c v + c 2 v c k v k + d w + d 2 w d m w m, which is a linear combination of vectors in S. Therefore, v + w span(s). We have therefore that span(s) is a non empty subset of R n which is closed under the vector space operations in R n ; that is, span(s) is a subspace of R n. Proposition Given a non empty subset, S, of R n, span(s) is the smallest subspace of R n which contains S; that is, is W is any subspace of R n such that S W, then span(s) W. Proof: Let V denote the smallest subspace of R n that contains S; that is, (i) V is a subspace of R n ; (ii) S V ; and (iii) for any subspace, W, of R n such that S W, V W. We show that V = span(s). By Proposition 2.5.4, span(s) is a subspace of R n. Observe also that S span(s),

29 2.6. FINITELY GENERATED SUBSPACES 29 since v S implies that v = v span(s). It then follows that V span(s), (2.7) since V is the smallest subset of R n which contains S. It remains to show then that span(s) V. Let v span(s); then, there exist vectors v, v 2,..., v k in S such that v = c v + c 2 v c k v k for some scalars c, c 2,..., c k. Now, since S V, v i V for all i =, 2,..., k. It then follows from the closure of V with respect to scalar multiplication that c i v i V for all i, 2,..., k. Applying the closure of V with respect to vector addition we then get that c v + c 2 v c k v k V ; that is v V. We have then shown that v span(s) v V ; that is, span(s) V. Combining this with (2.7), we conclude that span(s) = V ; that is, span(s) is the smallest subspace of R n which contains S. Remark (The Span of the Empty Set). In view of Proposition 2.5.5, it makes sense to define span( ) = {}. Indeed, {} is the smallest subset of R n and {}. 2.6 Finitely Generated Subspaces We have seen that for any subset, S, of R n, span(s) is a subspace of R n. If the set S is finite, we will say that span(s) is a finitely generate subspace of R n. Definition 2.6. (Finitely Generated Subspaces). A subspace, W, of R n is said to be finitely generate iff W = span(s) for some finite subset S of R n. Example Since {} = span( ), by definition, it follows that {} is finitely generated because is finite.

30 3 CHAPTER 2. EUCLIDEAN N DIMENSIONAL SPACE Example Let e, e 2,..., e n be vectors in R n given by e =., e 2 =.,..., e n =.. We show that R n = span{e, e 2,..., e n }. (2.8) This will prove that R n is finitely generated. To see why (2.8) is true, first observe that span{e, e 2,..., e n } R n. (2.9) x x 2 Next, let. denote any vector in Rn. We then have that x n x x 2. = x n x x. + x x n = x. + x x n. = x e + x 2 e x n e n, x 2 which shows that. is in the span of {e, e 2,, e n }. Thus, x n R n span{e, e 2,..., e n }. Combining this with (2.9) yields (2.8), which shows that R n is finitely generated. We will eventually show that all subspaces of R n are finitely generated. Before we do so, however, we need to make a short incursion into the theory of systems of liner equations.

31 2.7. CONNECTIONS WITH SYSTEMS OF LINEAR EQUATIONS Connections with the Theory of Systems Linear Equations We have seen that the questions of whether a given set of vectors in R m is linearly independent can be translated into question of whether a homogeneous system of the form a x + a 2 x a n x n = a 2 x + a 22 x a 2n x n =. =. a m x + a m2 x a mn x n =, (2.2) has only the trivial solution or many solutions. In this section we study these systems in more detail. In particular, we will see that in the case m < n, then the system (2.2) has infinitely many solutions. This result will imply that any set of n vectors in R m, where n > m, is linearly dependent. We will illustrate this with an example in R 2. Example Let v = ( 2), v 2 = set {v, v 2, v 3 } is linearly dependent. Solution: Consider the equation ( ) 2 and v 3 = ( ). Show that the c v + c 2 v 2 + c 3 v 3 =, (2.2) where denotes the zero vector in R 2 and c, c 2 and c 3 are scalars. This vector equation leads to the system of equations { c + 2c 2 + c 3 = (2.22) 2c c 2 + c 3 =. Solving for c in the first equation and substituting into the second equation leads to the system { c + 2c 2 + c 3 = (2.23) 5c 2 c 3 =. Observe that systems (2.22) and (2.23) have the same solutions since we simply solved for one of the variables in one equation and substituted into the other. Similarly, we can now solve for c 2 in the second equation in (2.23) and substitute for it in the first equation of the same system to get { c c 3 = (2.24) 5c 2 c 3 =. We can then solve for c and c 2 in system (2.24) to get { c = 3 5 c 3 c 2 = 5 c 3. (2.25)

32 32 CHAPTER 2. EUCLIDEAN N DIMENSIONAL SPACE The variables c and c 2 in system (2.24) are usually called the leading variables of the system; thus, the process of going from (2.24) to (2.25) is usually referred to as solving for the leading variables. System (2.25) shows that the leading variables, c and c 2, depend on c 3, which is arbitrary. We may therefore define c 3 = 5t, where t is an arbitrary parameter to get the solutions c = 3t c 2 = t c 3 = 5t, so that the solution spaces of system (2.22) is 3 W = span 2. 5 (2.26) We therefore conclude that the vector equation (2.2) has non trivial solutions and therefore {v, v 2, v 3 } is linearly dependent Elementary Row Operations The process of going from system (2.22) to the system in (2.24) can also be achieved by a procedure that uses elimination of variables instead of substitution. For instances, we can multiply the first equation in (2.22) by the scalar 2, adding to the second equation and replacing the second equation by the result leads to the system: { c + 2c 2 + c 3 = (2.27) 5c 2 c 3 =, which is the same system that we got in (2.24). This procedure does not change the solution space of the original system. In general, the solution space for the pair of equations { ai x + a i2 x a in x n = (2.28) a k x + a k2 x a kn x n = is the same as that of the pair { ai x + a i2 x a in x n = (ca i + a k )x + (ca i2 + a k2 )x (ca in + a kn )x n =, x 2 where c is any scalar. To see why this is so, let. (2.29); thus, from the second equation in the system, x x n (2.29) be a solution of system (ca i + a k )x + (ca i2 + a k2 )x (ca in + a kn )x n =.

33 2.7. CONNECTIONS WITH SYSTEMS OF LINEAR EQUATIONS 33 It then follows, using the distributive properties, that ca i x + a k x + ca i2 x 2 + a k2 x ca in x n + a kn x n =. Thus, by the associative properties and the distributive property again, c(a i x + a i2 x a in x n ) + a k x + a k2 x a kn x n =. x x 2 Consequently, since also satisfies the first equation in (2.29), we get that. x n a k x + a k2 x a kn x n =, which is the second equation in (2.28). Hence, x x 2. x n x x 2 system (2.28). A similar argument shows that if. x n is also a solution of is also a solution of system (2.28), then it is also a solution of system (2.29). Adding a scalar multiple of one equation to another equation and replacing the second equation by the resulting equation is an example of an elementary row operation. Other elementary row operations are: () multiply an equation by a no zero scalar and replace the equation by the result of the scalar multiple, and (2) swap two equations. It is clear that the later operation does not change the solution space of the system; in the former operation, since the scalar is non-zero, the solution space does not change either. To see why this is the case, x x 2 note that if. x n is a solution of then, since c, we see that c(a i x + a i2 x a in x n ) =, a i x + a i2 x a in x n =. We illustrate this by multiplying the second equation in (2.27) by /5 to get. { c + 2c 2 + c 3 = c c (2.3) 3 =,

34 34 CHAPTER 2. EUCLIDEAN N DIMENSIONAL SPACE The system in (2.3) is in what is known as row echelon form, in which the leading variables c and c 2 have as coefficient. We can perform a final row operation on the system in (2.3) by multiplying the second equation in the system by the scalar 2 and adding to the first equation to get { c c 3 = c c (2.3) 3 =. The system in (2.3) is said to be in reduced row echelon form. It can be solved for the leading variables to yield the system in (2.25) Gaussian Elimination Observe that in going from system (2.22) to system (2.3) by performing elementary row operations in the equations, as outlined in the previous section, the operations only affected the coefficients; the variables c, c 2 and c 3 acted as place holders. It makes sense, therefore, to consider the coefficients only in order to optimize calculations. The coefficients in each equation in system (2.22) can be represented as rows in an array of numbers shown in equation ( ) 2. (2.32) 2 The two dimensional array in (2.32) is known as the augmented matrix for the system (2.22). The elementary operations can then be performed on the rows of the augmented matrix in (2.32) (hence the name, elementary row operations). If we denote the rows first and second row in the matrix in (2.32) by R and R 2, respectively, we can denote and keep track of the row operations as follows: ( ) 2 2R + R 2 R 2 :. (2.33) 5 2R + R 2 R 2 in (2.33) indicates that we have multiplied the first row in (2.32) by 2, added the scalar product to the second, and replaced the second row by the result. The rest of the operations can be indicated as follows: ( ) 2 ( /5)R 2 R 2 :, (2.34) /5 and 2R 2 + R R : ( ) 3/5. (2.35) /5 The matrix in (2.34) is in row echelon form, and that in (2.35) is in reduced row echelon form. The process of going from an augmented matrix for a system to any of its row echelon forms by performing elementary row operations is known as Gaussian Elimination or Gauss Jordan reduction. We will present here two more examples in the context of determining whether a given set of vectors is linearly independent or not.

35 2.7. CONNECTIONS WITH SYSTEMS OF LINEAR EQUATIONS 35 Example Determine whether the set of vectors {v, v 2, v 3 } in R 3, where 2 v =, v 2 = 5 and v 3 = 4, 3 is linearly independent or not. Solution: Consider the equation c + c c 3 4 = 3. (2.36) This leads to the system c + 2c 2 = 5c 2 4c 3 = c + c 2 + 3c 3 =. (2.37) Starting with the augmented matrix 2 5 4, (2.38) 3 we perform the following elementary row operations on the matrix in (2.38): 2 5 4, R + R 3 R (/5)R 2 R 2 4/5, /5, 3R 2 + R 3 R 3 27/5 and (5/27)R 3 R 3 2 4/5 where we have indicated the row operation by the row on which the operation was performed. It then follows that the system in (2.37) is equivalent to the system c + 2c 2 = c 2 (4/5)c 3 = (2.39) c 3 =.,

36 36 CHAPTER 2. EUCLIDEAN N DIMENSIONAL SPACE System (2.39) is in row echelon form and can be solved to yield c 3 = c 2 = c =. Consequently, the vector equation (2.36) has only the trivial solution, and therefore the set {v, v 2, v 3 } is linearly independent. Example Determine whether the set of vectors {v, v 2, v 3 } in R 3, where v =, v 2 = 2 5 and v 3 = 5, 3 is linearly independent or not. Solution: Consider the equation c + c c 3 5 = 3. (2.4) This leads to the system c + 2c 2 = 5c 2 + 5c 3 = c + c 2 + 3c 3 =. (2.4) Starting with the augmented matrix 2 5 5, (2.42) 3 we perform the following elementary row operations on the matrix in (2.42): 2 (/5)R 2 R 2, R + R 3 R , 3R 2 + R 3 R 3 and 2R 2 + R R 2 We then conclude that the system (2.4) is equivalent to the system { c 2c 3 = c 2 + c 3 =,. (2.43)

37 2.7. CONNECTIONS WITH SYSTEMS OF LINEAR EQUATIONS 37 which is in reduced row echelon form and can be solved for the leading variables c and c 2 to yield c = 2t c 2 = t c 3 = t, where t is an arbitrary parameter. It then follows that the system in (2.4) has infinitely many solutions; consequently, the vector equation in (2.4) has nontrivial solutions and therefore the set {v, v 2, v 3 } is linearly dependent. Examples and illustrate what can happen in general when we are solving homogeneous linear systems: either () the system has only the trivial solution; or (2) the system has infinitely many solutions. In the case in which there are infinitely many solutions, the system is equivalent to one in which there are more unknowns than equations, as was the case in system (2.43) Example In the following section we prove that this is always the case: a homogenous systems with more unknowns than equations has infinitely many solutions The Fundamental Theorem of Homogenous Systems of Linear Equations The fundamental theorem for homogenous linear systems states that a homogenous system of more unknowns than equations has a nontrivial solutions. This is Theorem.5E in our text [TT7, pg. 6]. We present here slight variation of that theorem: Theorem (Fundamental Theorem of Homogeneous Linear Systems). A homogeneous system of m linear equations in n unknowns, a x + a 2 x a n x n = a 2 x + a 22 x a 2n x n =... a m x + a m2 x a mn x n =, with n > m has infinitely many solutions. (2.44) Proof: Perform Gauss Jordan reduction of the augmented matrix of the system in (2.44) to obtain the equivalent augmented matrix b 2 b 3 b k b,k+ b n b 23 b 2k b 2,k+ b 2n b k,k+ b kn in row echelon form, where k m. The previous system can be further reduced

38 38 CHAPTER 2. EUCLIDEAN N DIMENSIONAL SPACE to b,k+ b n b 2,k+ b 2n , b k,k+ b kn which leads to the system x + b,k+ x k+ + b,k+2 x b nx n = x 2 + b 2,k+ x k+ + + b 2n x n =... x k + b k,k+ x k+ + b k,n x n =, (2.45) where k m, equivalent to (2.44). We can solve for the leading variables, x, x 2,..., x k in (2.45) in terms of x k+,..., x n, which can be set to equal arbitrary parameters. Since n > m and k m, there are n k such parameters, It follows that system (2.45) has infinitely many solutions. Consequently, (2.44) has infinitely many solutions. A consequence of the Fundamental Theorem is the following Proposition which will play a crucial role in the study of subspaces of R n in the next section. Proposition Any set of vectors {v, v 2,..., v k } in R n with k > n must be linearly dependent. Proof: Consider the vector equation c v + c 2 v c k v k, =. (2.46) Since the set {v, v 2,..., v k } is a subset of R n, we can write v = a a 2 a 3. a n, v 2 = a 2 a 22 a 32. a n2,, v k = a k a 2k a 3k.. a nk Hence, the vector equation in (2.46) translate into the homogeneous system a c + a 2 c a k c k = a 2 c + a 22 c a 2k c k = (2.47)... a n c + a n2 n a nk c k =, of n linear equations in k unknowns. Since k > n, the homogenous system in (2.47) has more unknowns than equations. It then follows from the Fundamental Theorem that system (2.47) has infinitely many solutions. It then follows that the vector equation in (2.46) has a nontrivial solution, and therefore, by Proposition 2.4.5, the set {v, v 2,..., v k } is linearly dependent.

39 2.7. CONNECTIONS WITH SYSTEMS OF LINEAR EQUATIONS 39 Example By Proposition 2.7.5, the set S =, 2, 5, 5 4 is a linearly dependent subset of R 3. We will now show how to find a subset of S which is linearly independent and which also spans span(s). Solution: Denote the elements of S by v, v 2, v 3 and v 4, respectively, and consider the vector equation c v + c 2 v 2 + c 3 v 3 + c 4 v 4 =. (2.48) Since S is a linearly dependent, equation (2.48) has nontrivial solutions. Our goal now is to find those nontrivial solutions to obtain nontrivial linear relations between the elements of S which will allow us to express some of the vectors as linear combinations of the other ones. Those vectors in S which can be expressed as linear combinations of the others can be discarded. We perform this procedure until we find a linearly independent subset of S which which also spans span(s). Equation (2.48) leads to the system c + c 3 c 4 = c + 2c 2 + 5c 3 + 5c 4 = c + c 2 + c 3 + 4c 4 =, which has the augmented matrix R R R 3 4 (2.49) Performing the elementary row operations R + R 2 R 2 and R + R 3 R 3, we obtain the augmented matrix: Next, perform 2 R 2 R 2 and R 2 + R 3 R 3 in succession to obtain 2 3. Hence, the system in (2.57) is equivalent to the system { c + c 3 c 4 = c 2 + 2c 3 + 3c 4 =. (2.5)

40 4 CHAPTER 2. EUCLIDEAN N DIMENSIONAL SPACE Solving for the leading variables c and c 2 in (2.5) then yields the solutions c = t + s c 2 = 2t 3s (2.5) c 3 = t c 4 = s, where t and s are arbitrary parameters. Taking t = and s = in (2.5) yields the nontrivial linear relation v + 2v 2 v 3 =, by virtue of the vector equation (2.48), which shows that v 3 = v + 2v 2 and therefore v 3 span{v, v 2 }. (2.52) Similarly, taking t = and s = in (2.5) yields the nontrivial linear relation v 3v 2 + v 4 =, from which we get that v 4 = v + 3v 2, and therefore It follows from (2.52) and (2.53) that v 4 span{v, v 2 }. (2.53) {v, v 2, v 3, v 4 } span{v, v 2 }. Consequently, since span{v, v 2, v 3, v 4 } is the smallest subspace of R 3 contains {v, v 2, v 3, v 4 }, by Proposition 2.5.5, which span{v, v 2, v 3, v 4 } span{v, v 2 }. Combining this with we obtain that span{v, v 2 } span{v, v 2, v 3, v 4 }, span{v, v 2 } = span(s). It remains to check that {v, v 2 } is linearly independent. However, this follows from the fact that v and v 2 are not scalar multiples of each other Nonhomogeneous Systems Asking whether a vector v R n is in the span of the set {v, v 2,..., v k } in R n leads to the system of n linear equations in k unknowns a c + a 2 c a k c k = b a 2 c + a 22 c a 2k c k = b 2 (2.54)... a n c + a n2 c a nk c k = b n,

41 2.7. CONNECTIONS WITH SYSTEMS OF LINEAR EQUATIONS 4 and where v = a a 2 a 3. a n, v 2 = a 2 a 22 a 32. a n2 v =,, v k = b b 2 b 3.. b n a k a 2k a 3k,. a nk If v is not the zero vector in R n, then the system in (2.54) is a nonhomogeneous. In general, nonhomogeneous system might or might not have solutions. If they do have a solution, they either have exactly one solution or infinitely many solutions. We can analyze the system in (2.54) by considering the augmented matrix a a 2 a k b a 2 a 22 a 2k b a n a n2 a nk b n (2.55) and performing elementary row operations on the rows of the matrix in (2.55). Example Determine whether or not the vector 2, is in the span 3 of the set S =, 2, 5, 5. 4 Solution: Denote the elements of S by v, v 2, v 3 and v 4, respectively, and consider the vector equation where c v + c 2 v 2 + c 3 v 3 + c 4 v 4 = v, (2.56) v = 2. 3 Equation (2.56) leads to the system c + c 3 c 4 = c + 2c 2 + 5c 3 + 5c 4 = 2 c + c 2 + c 3 + 4c 4 = 3, (2.57)

42 42 CHAPTER 2. EUCLIDEAN N DIMENSIONAL SPACE which has the augmented matrix R R R Performing the elementary row operations R + R 2 R 2 and R + R 3 R 3, we obtain the augmented matrix: Next, perform 2 R 2 R 2 and R 2 + R 3 R 3 in succession to obtain 2 3 /2. 5/2 The third row in the previous matrix yields = 5/2, which is impossible. Therefore, the vector equation in (2.56) is not solvable. Hence, 2 is not 3 in the span of the set S. 2.8 Maximal Linearly Independent Subsets The goal of this section is to prove that every subspace, W, of R n is the span of a linearly independent subset, S. In other words, Theorem Let W be a subspace of R n. There exists a subset, S, of W such that (i) S is linearly independent, and (ii) W = span(s). In the proof of Theorem 2.8. we will use Proposition 2.7.5, which says that any set of vectors {v, v 2,..., v k } in R n with k > n must be linearly dependent, and the following Lemma Let S = {v, v 2,..., v k } be a linearly independent subset of R n. If v span(s), then the set is linearly independent. S {v} = {v, v 2,..., v k, v} Remark The set S {v} is called the union of the sets S and {v}.

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