# NOTES on LINEAR ALGEBRA 1

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1 School of Economics, Management and Statistics University of Bologna Academic Year 205/6 NOTES on LINEAR ALGEBRA for the students of Stats and Maths This is a modified version of the notes by Prof Laura Guidotti, translated by Alessandra Giovagnoli.

2 Contents The vector space R n. Definition of vector space The numerical space R n Addition Scalar multiplication Linear combinations Linear dependence Subspaces of R n The linear span Bases and dimension of subspaces of R n Matrices 6 2. The space R m n The rank of a matrix Matrix echelon form and rank computation Vector structure of R m n Sum of two matrices Scalar multiplication Matrix product Properties of the matrix product The transpose of a matrix The rank of the product of two matrices Square matrices and determinant The ring structure of R n n Definition of determinant Properties of determinant Finding the inverse of a non-singular matrix Linear Systems Linear equations Systems of linear equations Cramer systems The general case: Gauss method Linear systems and subspaces of R n The four fundamental subspaces of a matrix The Euclidean structure of R n The inner (dot) product Euclidean norm Orthogonal vectors Orthogonal complement Euclidean distance

3 6 Orthogonal projections and least square approximations Projection onto a -dimensional subspace Orthogonal and orthonormal basis Gram-Schmidt process Orthogonal projection onto a (general) subspace The least square solution The case of a system without solutions The case of a system with infinite many solutions Projection matrices Linear transformations General definition Linear transformations between R n and R m Linear isometry and orthogonal matrices Symmetric endomorphisms and symmetric matrices Scalar matrices and homotheties Eigenvalues and Eigenvectors The characteristic polynomial of a matrix Properties of eigenvalues and eigenspaces Diagonalizing a matrix Diagonalizing symmetric matrices or self-adjoint endomorphisms Real quadratic forms

4 The vector space R n Chapter. Definition of vector space A vector space V over the real numbers R is a collection of objects called vectors, which may be added together and multiplied by real numbers, called scalars. The operations of vector addition and scalar multiplication must satisfy the following requirements (axioms). Let u, v and w be arbitrary vectors in V, and α and β arbitrary scalars in R Commutativity of addition Associativity of addition u + v = v + u. (u + v) + w = u + (v + w) Identity element of addition the zero vector, such that There exists an element 0 V, called u + 0 = 0 + u = u for all u V. Inverse elements of addition For every u V, there exists an element u V, called the opposite vector of u, such that u + ( u) = ( u) + u = 0. Distributivity of scalar multiplication with respect to vector addition for all u, v V and for all α R α(u + v) = αu + αv Distributivity of scalar multiplication with respect to addition of real numbers (α + β)v = αv + βv for all v V and for all α, β R Compatibility of scalar multiplication with real multiplication for all v V and for all α, β R α(βv) = (αβ)v

5 Identity element of scalar multiplication v = v, for all v V, where R is the multiplicative identity. Straightforward consequences of the axioms are: α0 = 0 0v = 0 ( α)v = α( v) = (αv) αv = 0 either α = 0 or v = 0. The mathematical definition of vector space.is very general and includes a wide variety of mathematical objects (e.g. the set of all the real functions is a vector space over the reals) but we are going to deal just with one important type, namely the numerical vector spaces R n, for all n, and their subspaces.2 The numerical space R n Given the set of real numbers R and a natural number n, n, we write: R n = {(x, x 2,..., x n ), x i R, i =,, n} for the set of all ordered n tuples of real numbers. The elements of R n are called points. The origin O is the point (0, 0,..., 0). The space R n is subject to intuitive physical interpretations when n =, 2, 3. In detail:. R (i.e. R) can be thought of as an algebraic model for the set of points of a (straight) line; 2. R 2 can be thought of as an algebraic model for the set of points of a plane; 3. R 3 can be thought of as an algebraic model for the set of points of the physical 3-dimensional space where we live. These identifications depend on the choice of a (Cartesian) coordinate system, as we will see in the following. Geometric objects of the real line, plane or space are described by algebraic expressions and operations in R, R 2 and R 3 respectively. For n 4 there is no physical-geometrical interpretation, but R n is a useful mathematical model to study a variety of different real life phenomena that depend on a large number of variables. 2

6 Now we come to vectors. In R 3 (also R, R 2 ) a vector can be thought of as an action that shifts points in space. All the points are shifted in the same direction and verse and by the same distance. We can identify each vector with the point where the origin O is shifted to. Example... Given a shift of all the points of R 2 such that the origin (0, 0) is shifted to point (, 3), we describe this by a vector v = (, 3). The same vector v = (, 3) will shift point (, 2) to (0, 5). 2. In R 3, (, 2, 4) is mapped to (6, 5, 3) by the vector (5, 7, ). We need to extend this idea to R n n. In R n if O is mapped to (x, x 2,..., x n ) by the vector v, we write v = (x, x 2,..., x n ). Vectors are usually indicated by letters in bold, e.g. v, or underlined, namely v, or sometimes with a little arrow over them, like this: v. The numbers x, x 2,..., x n are called the components or coordinates of the vector. Point (x, x 2,..., x n ) is called the position point of v. If a vector is identified by the action of shifting point A to point B, (respectively, the initial and end point of the vector) we write AB. If point (x, x 2,..., x n ) is shifted to point (y, y 2,..., y n ) by a vector u, then the components of u are (y x, y 2 x 2,..., y n x n ). Two vectors u = (x, x 2,..., x n ), v = (y, y 2,..., y n ) are equal if and only if x i = y i, i =,..., n. Now we can say that R n = {(x, x 2,..., x n ), x i R, i =,, n} is also a set of vectors. It may be confusing to have the same notation for points and vectors. Later, we shall see that it is more useful to write vectors as vertical n-tuples: x x 2... x n or x x 2... x n, but in this chapter we stick to the horizontal notation. Particularly. R can be thought of as an algebraic model also for the set of geometric vectors belonging to the same line and having the same initial point; 2. R 2 can be thought of as an algebraic model also for the set of geometric vectors whose directions lie on a plane and have the same initial point (of the plane); 3. R 3 can be thought of as an algebraic model also for the set of all geometric vectors of the physical 3-dimensional space having the same initial point. 3

7 In order to be able to say that R n is a vector space over R operations must be defined between vectors and between vectors and scalars of R, namely addition of two vectors and multiplication of a vector by a scalar, and they must satisfy the axioms of Section...2. Addition Take any two vectors u = (x, x 2,..., x n ) and v = (y, y 2,..., y n ) in R n, which we can think of as acting on the points, and imagine performing both actions (shifts) one after the other. For instance given u = (3, 2) and v = (, 4) in R 2, first the origin is mapped to (3, 2) by u, and then point (3, 2) is mapped to point (4, 2) by v. As a combined result we get that the origin is mapped to point (4, 2) by a new vector w. We call this combination of vectors their addition. The vector w that performs the combined shift is called the sum of u and v and is denoted by u + v. Definition.. for all u, v R n. u + v = (x + y, x 2 + y 2,..., x n + y n ), Namely, the sum of two vectors is the vector whose components are the sums of the components with the same subscripts. In the example above, w = (3 +, 2 4) = (4, 2). Properties of the addition in R n : Commutative property. This property is satisfied because for all u, v R n. u + v = (x + y, x 2 + y 2,..., x n + y n ) = (y + x, y 2 + x 2,..., y n + x n ) = v + u. Associative property This property is satisfied because (u + v) + w = ((x + y ) + w, (x 2 + y 2 ) + w 2,..., (x n + y n ) + w n ) for all u, v, w R n = (x + (y + w ), x 2 + (y 2 + w 2 ),..., x n + (y n + w n )) = u + (v + w) The zero vector property This property is satisfied because 0 = (0, 0,, 0) is such that u + 0 = 0 + u = u for all u R n. 4

8 The opposite vector property. This property is satisfied because for any u = (x, x 2,..., x n ) R n the vector u = ( x, x 2,..., x n ) is such that u + ( u) = ( u) + u = Scalar multiplication Definition.2. For all vectors v R n and real numbers α R, define Properties αv = α(x, x 2,..., x n ) = (αx, αx 2,..., αx n ). Distributivity of scalar multiplication with respect to vector addition. This property is satisfied because α(u + v) = α(x + y, x 2 + y 2,..., x n + y n ) = (αx + αy, αx 2 + αy 2,..., αx n + αy n ) = αu + αv Distributivity of scalar multiplication with respect to addition of real numbers. This property is satisfied because (α + β)v = ((α + β)y, (α + β)y 2,..., (α + β)y n ) = α(y, y 2,..., y n ) + β(y, y 2,..., y n ) = αv + βv Compatibility of scalar multiplication with real multiplication This property is satisfied because α(βv) = α(βy, βy 2,..., βy n ) = (αβy, αβy 2,..., αβy n ) = (αβ)v Identity element of scalar multiplication. This property is satisfied because v = (y, y 2,..., y n ) = v where R is the multiplicative identity. For all v, w R n and for all α, β R Because of the above operations (addition and multiplication by a scalar) and their properties, we can say that R n is a vector space over R. Let us look at some of its main features. 5

9 .2.3 Linear combinations We are going to combine the above operations into one. Definition.3. Given the vectors v, v 2,, v r, and the scalars α, α 2,, α r, combining addition and scalar multiplication we get the vector: v = α v + α 2 v α r v r which is called a linear combination of v, v 2,..., v r with coefficients (or weights) α, α 2,, α r. Example.2.. In R 4 the linear combination of vectors v = (, 0, 2, 0), v 2 = (3, 4,, 2), v 3 = (0, 2, 0, ) with coefficients λ =, λ 2 = 6, λ 3 = 2 is: v + 6v 2 + 2v 3 = (7, 28, 4, 4). 2. In R 2, the linear combination of vectors v = (, 3) and w = (3, 9) with weights λ = 3, µ = is: 3v + w = ( 3 + 3, 9 + 9) = (0, 0) = In R 3, we can ask ourselves whether vector u = (2,, 0) is or is not a linear combination of the vectors v = (, 3, 0) and w = (0,, 0), i.e. whether there exist two scalars α and β such that: namely u = αv + βw (2,, 0) = α(, 3, 0)+β(0,, 0) = (α, 3α, 0)+(0, β, 0) = (α, 3α+β, 0) hence α = 2, β = 5. Thus vector u is indeed a linear combination of v and w : u = 2v + 5w. It is of interest to point out that v too is a linear combination of u and w, since from u = 2v + 5w, one obtains v = 2 u 5 2 w, and also that w is a linear combination of u and v : w = 5 u 2 5 v. 4. Every vector of R 3 is a linear combination of the three vectors e = (, 0, 0), e 2 = (0,, 0), e 3 = (0, 0, ), since v = (x, x 2, x 3 ) = (x, 0, 0) + (0, x 2, 0) + (0, 0, x 3 ) = = x (, 0, 0) + x 2 (0,, 0) + x 3 (0, 0, ) = = x e + x 2 e 2 + x 3 e 3. 6

10 More in general, every vector of R n is a linear combination of the n vectors: e = (, 0,, 0), e 2 = (0,,, 0),, e n = (0, 0,, ) as follows: v = (x, x 2,..., x n ) = x e + x 2 e x n e n. Given a set of vectors in R n, {v, v 2,..., v r }, the linear combination with all the coefficients equal to zero is the zero (or null) vector 0: 0v + 0v v r = 0. This is called the trivial linear combination..2.4 Linear dependence Definition.4. Given r vectors of R n a, a 2,, a r the vectors of the set {a, a 2,, a r } are said to be linearly dependent if there are scalars λ, λ 2,, λ r, not all zero such that: λ a + λ 2 a λ r a r = 0 i.e. there exists a non-trivial linear combination which gives the zero vector. A set of vectors are linearly independent when they are not linearly dependent, i.e. the only linear combination of all the vectors of the set giving the zero vector is the trivial one. Example.3.. In R 3 the vectors a = (,, 0), a 2 = (2, 0, ), a 3 = (, 3, 2) are linearly dependent since 3a 2a 2 a 3 = In R 4 the vectors of the set : B = {v = (, 0, 2, ), v 2 = (0,,, 0), v 3 = (,,, )} are linearly independent. To show this, write λ v +λ 2 v 2 +λ 3 v 3 = (λ +λ 3, λ 2 +λ 3, 2λ +λ 2 +λ 3, λ +λ 3 ) = (0, 0, 0, 0) 7

11 We need to solve a set of linear equations in the unknowns λ, λ 2, λ 3 λ + λ 3 = 0 λ 2 + λ 3 = 0 2λ + λ 2 + λ 3 = 0 λ + λ 3 = 0 equivalent to: λ = 0 λ 2 = 0 λ 3 = 0 So the only linear combination is the one with coefficients equal to zero. 3. In R n the set of vectors {e = (, 0,, 0), e 2 = (0,,, 0),, e n = (0, 0,, )} is linearly independent: λ e +λ 2 e 2 + +λ n e n = (λ, λ 2,, λ n ) = 0 λ = λ 2 = = λ n = If a set contains just one vector v, from: αv = 0 α = 0 or v = 0 there follows that the set {v} is linearly dependent if and only if v = 0. Proposition.. If a set A = {a, a 2,, a r } contains at least two vectors, the vectors of A are linearly dependent if and only if at least one of them is a linear combination of the others. Proof If λ a + λ 2 a λ r a r = 0 and, say, λ r 0, then vector a r is a linear combination of the first r vectors of A, since Conversely if the equality a r = λ λ r a λ 2 λ r a 2 + λ r λ r a r. a r = µ a + µ 2 a µ r a r, µ a + µ 2 a µ r a r a r = 0 shows a non-trivial linear combination (the coefficient of vector a r is 0) of the vectors of A. In particular two vectors v, w are linearly dependent if and only if v = αw; in short, if the two vectors are proportional (also called parallel). The following is an important result 8

12 Proposition.2. Let A = {a, a 2,, a r } and A {0}, then it is possible to find a maximal independent subset B A namely a subset B such that the vectors of B are linearly independent; if a i A and a i / B, the vectors of B {a i } are linearly dependent. Example.4.. In R 4 we look for a maximal independent subset of A = {a = (, 0,, 0), a 2 = (0,, 2, ), a 3 = (0, 0, 0, 0), a 4 = (,, 3, ), a 5 = (, 0, 0, 0)}; A is a dependent set, since it contains the zero vector. The vector a is linearly independent because it is non-zero; the vectors {a, a 2 } are independent, because they are not proportional; the vectors {a, a 2, a 3 } are linearly dependent. Also {a, a 2, a 4 } are linearly dependent because a 4 = (,, 3, ) = a + a 2 ; lastly, B = {a, a 2, a 5 } is a set of linearly independent vectors. Starting from a vector different from a, one obtains a maximal independent set different from B, for instance C = {a 2, a 4, a 5 }. 2. What about E = {e = (, 0, 0), e 2 = (0,, 0), e 3 = (0, 0, )} in R 3? Are they independent? What is a maximal independent subset?.3 Subspaces of R n In the vector space R n we are going to consider other vector spaces that are contained in it. Definition.5. We say that a non empty subset S R n is a vector subspace if the following holds: for all v, v 2 S v + v 2 S, for all α R, and for all v S αv S. In other words, a subset of vector space R n is a subspace if it contains all the linear combinations of its vectors. It can be shown that all the properties of a vector space still hold in a vector subspace too. Special subspaces of R n are i) {0}, the null subspace, containing only the zero vector; ii) R n itself; iii) the subspace of all the vectors proportional to a given non-null one: {λv 0, λ R}. In the physical 3 dimensional space this subspace corresponds to a line through the origin in the direction of v 0 ; 9

13 iv) subsets consisting of all the vectors with certain coordinates equal to zero, e.g. those for which the second coordinate is 0, like: {v = (x, 0, z); x, z R}. In the physical 3 dimensional space with Cartesian coordinates this subspace corresponds to the xz-plane. Example.5.. Consider the following subsets of R 2 : S = {v = (x, 2x); x R} T = {v = (x, ); x R} W = {v = (0, y); y R}. S is a subspace (see iii) above), since given any two vectors of S, (x, 2x) and (x, 2x ), their sum (x, 2x) + (x, 2x ) = (x + x, 2(x + x )) belongs to S, and so does the product α(x, 2x) = (αx, 2αx). The set T is not a subspace of R 2, since (x, ) + (x, ) = (x + x, 2) is not a vector of T. The set W is a subspace of R 2 (see iv) above). 2. Consider the following subsets of R 4 : S = {u = (x, 2x, 0, 0); x R} T = {v = (α, β, α + β, 2β); α, β R} W = {w = (α, α + 3, α, 0); α R}. S and T are subspaces of R 4, W is not, since it is not closed with respect to either addition or scalar multiplication, meaning that the sum of two vectors of W is not necessarily a vector of W, nor is the product of a vector of W by a scalar. Remark If S is a subspace of R n then it contains the zero vector: to see this take a vector v S and 0 R, then 0v = 0 S. The converse statement is not true: for example the set W = {(x, y) R 2 y = x 2 } R 2 contains (0, 0), but it is not a subspace. In fact (, ), (2, 4) W but (, ) + (2, 4) = (3, 5) / W. Definition.6. The intersection S S 2 of two vector subspaces S and S 2 of R n is the set of all the vectors that belong to both: S S 2 = {u such that u S, u S 2 }. It is never empty, since it contains at least the zero vector. It is straightforward to check that: Proposition.3. For any two vector subspaces S and S 2 of R n, S S 2 is a vector subspace of R n. 0

14 On the contrary, the union of two vector subspaces, S S 2, is not in general a subspace. However, we can talk about the sum of two subspaces. Definition.7. For any two vector subspaces S and S 2 of R n, define their sum S + S 2 to be the set of all the sums u + v of a vector u S and a vector v S 2 : S + S 2 = {u + v, for all u S, v S 2 }. When S S 2 = {0} the sum of two subspaces is called a direct sum, denoted as S S 2. It is also straightforward to check that: Proposition.4. For any two vector subspaces S and S 2 of R n, S + S 2 is a vector subspace of R n. Example.6.. Consider the following subspaces of R 4 : S = {u = (γ, 2γ, 0, 0); γ R}, T = {v = (0, α, β, α + β); α, β R} Their intersection S T is {0} and the sum is S T = {(γ, α + 2γ, β, α + β) α, β, γ R}. This is not the whole space R 4, since for instance (0,, 0, ) R 4 but (0,, 0, ) / S T. 2. The following subspaces of R 3 S = {u = (γ, γ, δ); γ, δ R}, S 2 = {v = (α, β, α); α, β R} are such that S S 2 = {w = (α, α, α); α R} and S + S 2 = R 3, since any vector (x, y, z) of R 3 can be written as (x, y, z) = ( x 2, x 2, z x 2 ) + ( x 2, y x 2, x 2 )..4 The linear span We define the subspace spanned (i.e. generated) by a finite set A = {a, a 2,, a r } of vectors of R n, which we denote as span(a): span(a) is the set of all linear combinations of vectors of A: span(a) = {λ a + λ 2 a λ r a r, λ i R, i =,..., r}.

15 This set is closed with respect to addition and scalar multiplication in R n, since given any two linear combinations v = r i= λ ia i and w = r i= µ ia i v + w = r λ i a i + i= r µ i a i = i= r (λ i + µ i )a i which is still a linear combination of {a, a 2,, a r } so it still belongs to span(a). Similarly, for all α R: α r λ i a i = i= i= r (αλ i )a i which is a vector of span(a). Thus span(a) is a subspace of R n.. In R 3 the set {e = (, 0, 0), e 2 = (0,, 0)} spans the sub- Example.7. space i= span(e, e 2 ) = {λ e + λ 2 e 2 = (λ, λ 2, 0), λ, λ 2 R}. 2. In R 2, the vector v = (, ) spans the subspace: span(v) = {λv = (λ, λ), λ R} which geometrically consists of all the vectors belonging to the line y = x. 3. The null subspace {0} of R n is spanned by the null vector The whole space R n is spanned by the set: since E = {e = (, 0,, 0), e 2 = (0,,, 0),, e n = (0, 0,..., )} v = (x, x 2,..., x n ) = x e + x 2 e x n e n. The following is an important property of R n and of all its subspaces. Proposition.5. Every subspace S R n is finitely generated, i.e. all the vectors of S can be expressed as linear combinations of a finite number of vectors of S. This is not true of all vector spaces: the one consisting of all the real functions in NOT finitely generated..5 Bases and dimension of subspaces of R n Suppose S is a subspace of R n spanned by the set A = {a, a 2,, a r }: S = span(a). We may ask ourselves whether all the vectors of A are essential in order to span S. This is related to whether or not they are dependent. 2

16 Theorem.. Let A = {a, a 2,, a r } R n, and let B be a linearly independent maximal subset of A. Then A and B span the same subspace. Definition.8. Let S R n be a non-null subspace of R n. We say that the set of vectors B = {b, b 2,, b r } S is a basis of S if: B spans S: S = span(b); the vectors of B are linearly independent. In other words, a set of vectors B = {b, b 2,, b r } forms a basis of a subspace S if and only if all the vectors of S can be written in a unique way as linear combinations of the vectors b, b 2,, b r.. Example.8.. The space R 3 is spanned by E = {e = (, 0, 0), e 2 = (0,, 0), e 3 = (0, 0, )}, This set is not redundant;the exclusion of one generator, e.g. e 3, gives the set F = {e = (, 0, 0), e 2 = (0,, 0)} and span(f ) = {αe + βe 2 = α(0,, 0) + β(0,, 0); α, β R} = {(α, β, 0); α, β R} which is a subspace of R 3 that does not contain all the vectors of the space R 3 ; for instance the vector e 3 and all its multiples are missing. More in general, in R n the set E = {e = (, 0,, 0), e 2 = (0,,, 0),, e n = (0, 0,, )} spans the whole space R n and is linearly independent, thus it is a basis of the vector space R n, called the canonical or standard basis of R n. 2. In R 4 the vectors a = (, 0,, ), a 2 = (0,,, ), a 3 = (2,,, ) are linearly independent, hence they form a basis of the subspace spanned by them S = span{a, a 2, a 3 }. 3. In R 3 the subspace T spanned by a = (, 0, ), a 2 = (2, 0, 2), a 3 = (, 0, ), a 4 = (0, 0, 0) has a basis consisting of just one vector, for instance B = {a = (, 0, )}. Another basis of T is {a 2 }, and so is {a 3 }, which shows that bases of subspaces are not unique. 3

17 Remark Every non-null subspace S of R n (including R n itself), has a basis. A fundamental property is the following. Theorem.2. Two different bases of the same subspace S always have the same number of elements. Definition.9. The number of vectors forming a basis of a subspace S is called the dimension of S, denoted dim(s) or dim S. The dimension of the null subspace is taken to be zero. Thus the dimension of a vector subspace S R n is the maximum number of independent vectors in S. The dimension of a subspace of R n is also the minimum number of vectors in a spanning set. Remark If S is a subspace of R n of dimension r, then 0 r n. Going back to the previous examples:. The vector space R n has dimension n, since the standard basis: E = {e = (, 0,, 0), e 2 = (0,,, 0),, e n = (0, 0,, )} has n elements. 2. The subspace S of R 4 spanned by the vectors: b = (, 0,, ), b 2 = (0,,, ), b 3 = (2,,, ) has dimension 3, since these vectors are independent and thus form a basis of S. 3. The subspace T of R 3 spanned by a = (, 0, ), a 2 = (2, 0, 2), a 3 = (, 0, ), a 4 = (0, 0, 0) has dimension, i.e. all its bases consist of just one vector. Example We want to find out whether the vectors s = (,, 0), s 2 = (, 0, ), s 3 = (2, 3, 0) are a basis of R 3. It suffices to check that they are linearly independent. 5. Is the set K = {v = (,,, 0), v 2 = (0,, 0, ), v 3 = (, 2, 0, ), v 4 = (, 2,, )} 4

18 a basis of R 4? Since we have four vectors in a 4-dimensional space, they form a basis of R 4 if they are linearly independent. It is easy to check that v 4 = v + v 2 + 0v 3 thus K is not a basis of R 4. However, K spans a subspace, W = span(k). What is the dimension of W? We look for a maximal independent subset of K. For instance K = {v = (,,, 0), v 2 = (0,, 0, ), v 3 = (, 2, 0, )}. Then W = span(k) = span(k ), and thus W has dimension 3. Remark The vector space R n contains subspaces of any dimension r, with r = 0,,..., n, n. For r = 0 take the null subspace. Otherwise, if r, starting from the canonical basis of R n, E = {e = (, 0,, 0), e 2 = (0,,, 0),, e n = (0, 0,, )}, we can build subspaces of any dimension, 2,..., n, n. We take span({e }), which has dimension, span({e, e 2 }) with dimension 2,..., span({e,..., e n }) with dimension n, while span(e) = R n has dimension n. One-dimensional subspaces of R n are called lines through the origin, 2- dimensional subspaces are called planes through the origin, the (n )-dimensional ones are the hyperplanes through the origin of R n. Remark Given two subspaces S, T of R n such that: and then: S T dim S = dim T, S = T. This is due to the fact that any basis of S is also a basis of T, hence the two subspaces are spanned by the same set. The following remarkable result holds true: Theorem.3 (Grassmann s Theorem). Let S and T be two subspaces of R n then dim(s + T) + dim(s T) = dim(s) + dim(t). 5

19 Chapter 2 2 Matrices 2. The space R m n Given the natural numbers m and n, (m, n ), consider mn real numbers, displayed in an m n array a a 2... a n a 2 a a 2n A =..... a m a m2... a mn This is referred to as a matrix with m rows and n columns, also described as an m n matrix, with elements (or entries) in R. An alternative notation uses square brackets instead of round ones a a 2... a n a 2 a a 2n A =..... a m a m2... a mn Each entry of the matrix is identified by two subscripts, the first one indicates the row and the second one the column: entry a ij is the one at the intersection of row i with column j. For instance [ ] is a 2 4 matrix where a 3 = 2. A matrix A can also be denoted as for short. For fixed m ed n, we denote with A = [a ij ], i =,..., m, j =,..., n R m n = {A = [a ij ] a ij R, i =,..., m, j =,..., n, }. the set of all the matrices with m rows and n columns with R coefficients. Two matrices A = [a ij ], B = [a ij ] R m n are equal if a ij = b ij, for each i =,..., m and j =,..., n. When m = n the matrix is called a square matrix, otherwise a rectangular matrix. The number of row (that equals the number of columns) of a square 6

20 matrix is called the order of the matrix. In a square matrix of order n, the n tuple (a, a 22,..., a nn ) is called the main diagonal of A a a 2... a n a 2 a a 2n A = a n a n2... a nn Given a square matrix A = [a ij ] R n n, we say that A is - upper triangular if all the entries below the main diagonal are zero, that is a ij = 0 if i > j; - lower triangular if all the entries above the main diagonal are zero, that is a ij = 0 if i < j - diagonal if all the entries off the main diagonal are zero, that is a ij = 0 if i = j (notice that a diagonal matrix is both upper and lower triangular). u u 2... u n l d u u 2n U = L = l 2 l D = 0 d u nn l n l n2... l nn d nn Example A = is a matrix with 3 rows and 4 columns C = is a 3 3 square matrix and its main diagonal is the 0 π vector ( 3, 0, π). A n matrix, i.e. a matrix with just one row, is called a row matrix. For instance K = [ ], which has row and 6 columns, is a row matrix. Similarly, an m matrix, i.e. a matrix with just one column, like for instance 7 H = 0 5 7

21 with 5 rows and column, is called a column matrix. Row and column matrices can be identified with the corresponding vectors e.g. matrix K can be identified with vector (, 0, 3, 0, 2, 9) while matrix H can be identificated with vector (, 7,, 0, 5). In other words, we have a natural bijection between the sets R n, R n and R n. 2.2 The rank of a matrix Let A R mxn be a matrix with m rows and n columns. The rows r,..., r m of A are m vectors of the vector space R n. This set spans a subspace of R n called the row space of matrix A indicated as R(A). The dimension of R(A) is called the row rank of A and is, by definition, the maximum number of independent rows of A. Clearly this dimension cannot be greater than the number of rows, i.e. m, nor can it be greater than n, since R(A) R n thus 0 dim R(A) min (m, n). The same applies to the columns of A. They are n vectors c, c 2,..., c n R m. The subspace spanned by the columns of A is called the column space of A, indicated as C(A). The column rank of A, namely the dimension of C(A), is such that Example 2.2. Let the rows space of A is the subspace 0 dim C(A) min(m, n). 0 0 A = R 3 4, R(A) = span {r = (, 0,, 0), r 2 = (0, 2,, 3), r 3 = (, 4, 3, 6)} R 4. The rows are linearly dependent since r 3 = r + 2r 2. A basis of R(A) is thus {r, r 2 } since r and r 2 are not proportional. So the row rank of A, that is dim R(A), is 2. The columns of matrix A are vectors of R 3 and C(A) = span c = 0, c 2 = 0 2, c 3 =, c 4 = 0 3 R Also the columns are linearly dependent, since c 3 = c 2 c 2 and c 4 = 3 2 c 2. So C(A) = span(c, c 2, c 3, c 4 ) = span(c, c 2 ). So {c, c 2 } is a basis for C(A), since c, c 2 are linearly independent, and also the column rank of A is 2. 8

22 In this example the row rank and the column rank of A coincide. This is in fact a general result, in any matrix the maximum number of linearly independent rows equals the maximum number of linearly independent columns. Theorem 2.. Let A R mxn, and let C(A) R m and R(A) R n be the column space and the row space of A respectively, then dim C(A) = dim R(A) Since the row rank of every matrix A is the same as the column rank, we will simply refer to it as the rank of the matrix A. That is Clearly, if A R m n then rank(a) = dim R(A) = dim C(A). 0 rank(a) min(m, n). Finding the rank of a matrix it is not an easy problem and requires ad hoc methods as we will see in the next subsection. However, for some special matrices it is easy. For instance, it is not difficult to check that for a diagonal matrix the rank is the number of non-zero elements on the main diagonal. As a consequence, for each r = 0,,..., n there exists an order n matrix of rank r. Definition 2.. An order n matrix A R n n is said to be of full rank or regular or non-singular if it has rank n; if the rank of A is less than n we call it singular. 2.3 Matrix echelon form and rank computation A row in a matrix is said to be a zero row if it has all zero entries (i.e. it is the zero vector). Definition 2.2. A matrix A is said to be in (row) echelon form if - all nonzero rows are above all zero ones and - the leading entry (that is the first non zero entry from the left) of a non zero row is always strictly to the right of the leading entry of the row above it. The leading entries of a matrix in (row) echelon form are also called pivots. Example 2.3. The matrices π and ( )

23 are in echelon form and the pivots are the squared entries. On the contrary, the matrices ( ) π and are not in echelon form. Clearly the number of pivots of a matrix in echelon form equals the number of nonzero rows. Moreover, for matrices in echelon form the rank is really easy to compute. Proposition 2.. The rank of a matrix in echelon form equals the number of nonzero rows. Being in echelon form is a very special condition, but each matrix can be transformed to echelon form by operating on its rows. We call elementary row operations the following operations performed on the rows of a matrix - row switching: a row is switched with another r i r j - scalar multiplication: each element in a row is multiplied by a nonzero scalar r i αr i - row combination: a row is replaced by the sum of itself and the multiple of another row r i r i + αr j. We say that two matrices are row equivalent if it is possible to change one to another by a finite sequence of elementary row operations. Theorem 2.2. Each matrix can be transformed to row echelon form by means of a finite sequence of elementary row operations, that is each matrix is row equivalent to a matrix in echelon form. A key observation for constructing an algorithm to transform to echelon form, is that the two conditions of Definition 2. imply that all entries in a column below a pivot are zeros. So, starting with a fixed matrix, we can transform it to echelon form as follows. among the nonzero entries choose one of the leftmost and switch its row with the first one 2. combining scalar multiplication and row combination, reduce to zero all the entries below the chosen element 3. consider the matrix obtained by forgetting about the first row and repeat steps. and 2. Generally there are infinite many different matrices in echelon form all equivalent to a given one. The important fact is that, since elementary operation do not change the row space, they all have the same rank. 20

24 Proposition 2.2. If A is row equivalent to B then R(A) = R(B) and so rank(a) = rank(b). Thus, to compute the rank of a given matrix, first we use elementary operations to transform to echelon form and then we count the number of pivots of the echelon form. In a similar fashion, it is possible to give the definition of column echelon form and elementary column operations. Moreover, all the results presented in this section still hold once we replace the word row with the word column. 2.4 Vector structure of R m n We define an operation of addition of two matrices, and another operation of multiplication of a matrix by a scalar Sum of two matrices In the set R m n of all m n matrices the sum of two matrices A = [a ij ] and B = [b ij ] is defined entrywise, i.e. A + B = [a ij + b ij ], namely the sum of A and B is the matrix whose entries are the sums of the corresponding entries in A and B Example 2.4. If A = and B = 5, their sum is the matrix C = A + B = Properties of the matrix sum in R m n : Commutative property for all A, B R m n. A + B = B + A Associative property for all A, B, C R m n. (A + B) + C = A + (B + C) 2

25 Identity element The matrix with all its entries equal to zero is called the null matrix and is denoted as O m n ; the null matrix has no effect on the sum, i.e. A + O m n = O m n + A = A for all A R m n. Opposite elements For all A = [a ij ] the matrix A = [ a ij ] is such that Scalar multiplication A + ( A) = ( A) + A = O m n. We can also multiply a matrix by a real number: if A = [a ij ] and α R and the following properties hold true: αa = [αa ij ] α(a + B) = αa + αb (α + β)a = αa + βa α(βa) = (αβ)a A = A for all A, B R m n and for all α, β R. In conclusion, the set of all matrices R m n is a vector space on the real numbers. The terminology of Chapter can be used for matrices as well, e.g. linear combinations of matrices, linearly independent matrices, spans, subspaces of R m n and so on. Example [ ] [ ] 2 2 = In R 2 2 the linear combination of the matrices [ ] [ ] [ ] A =, B =, C = with weights α = 3, β = 2, γ = is 3A 2B + C = [ ]

26 3. In R n n the matrices with only one element equal to and all the others equal to 0 are linearly independent and they span the whole space R n n. For instance, R 2 2 is spanned by the matrices since [ ] a a 2 a 2 a 22 [ ] 0 0 0, [ ] 0 0 0, [ ] 0 0 0, [ ] [ ] [ ] [ ] [ ] = a + a a a More in general R m n is spanned by the set of mn matrices having just one element equal to and all the others equal to 0, which are clearly linearly independent. Thus the dimension of R m n is mn. 4. The set D of all n n diagonal matrices is a subspace of the vector space R n n. D is spanned by the linearly independent matrices E =.. 0., E =.. 0.,..., E n = since for all d, d 22,..., d nn R, d d D =.. 0. = d E + d 22 E d nn E n d nn so the dimension of D is n. 2.5 Matrix product As we will see, one of the main feature of matrix theory is the possibility of identifying an m n matrix A it with a (linear) function from the vector space R n to the vector space R m. Given two matrices A = [a ij ] R m n and B = [b jk ] R n p, we can think of B as a function from the vector space R p to the vector space R n. Now think of applying both functions in succession, namely vectors go from R p to R n by means of B, and then from R n to the vector space R m by means of A. Can we do this in one step instead of two, by means of just one matrix C? If the answer is yes, C will have to be an m p matrix. It is indeed possible to define such a matrix C, which is called the row-bycolumn product of A and B, and is denoted C = AB. 23

27 The definition of C is based on the definition of dot or inner product of two x = (x,..., x n ) and y = (y,..., y n ) vectors of R n x, y = x y = x y + + x n y n = n x j y j. The rows of A have n components, i.e. are vectors of R n, and so are the columns of B. Then we can take the dot product of each row of A by each column of B. In detail, the dot product of a i the row i of A by b k, the column k of B, where i =,..., m and k =,..., p, is n c ik = a i, b k = a ij b jk. j= The matrix C = (c ik ) R m p is the row-by-column product AB. [ ] Example Given the matrices A = R and B = R 3 3, [ ] the row-by-column product of A by B is AB = R Note that the row-by-column product BA is undefined, since the rows of B are vectors of R 3 while the columns of A are vectors of R 2. [ ] The row-by-column product of A = by B = is doable. Taking the inner product of each row of A by each column of B, the result is: [ ] 0 2 C = AB = R In this case it is also possible to multiply B by A, since the rows of B and the columns of A are vectors of R 2 : BA = R The two products are clearly very different! [ ] [ ] Now take the matrices H = and K = ; we can compute both products HK and KH: [ ] 5 2 HK = 2 3 [ ] 4 KH = 2 and HK KH. i= 24

28 The three examples above show that the row-column product of two matrices A and B in general does not satisfy the commutative property. It may so happen, however, that AB = BA for some particular pairs of matrices A and B; in which case we say that the matrices commute. For instance, for D = [ ] [ ] 0, D 2 = we get 0 4 [ ] 2 0 D D 2 = D 2 D =. 0 2 This is in fact a general result: all diagonal matrices of the same size commute, since given a b a D =......, D 0 b = a n b n then a b b a a 2 b D D 2 = = 0 b 2 a = D 2D a n b n b n a n 2.6 Properties of the matrix product Associative property Suppose A is m n, B is n s and C is s t. Then AB is m s and BC is n t, so both products (AB) by C and A by (BC) are possible, both giving rise to an m t matrix. The following property holds Left distributive property (AB)C = A(BC) Suppose A is m n, B and C are both n s. Then B + C is n t, so the products A by B, A by C, and A by (B + C) are all possible, all giving rise to m s matrices. The following property holds Right distributive property A(B + C) = AB + AC, Suppose A and B are both m n, and C is n s. Then A + B is m n, so the products A by C, B by C, and (A + B) by C are all possible, all giving rise to m s matrices. The following property holds (A + B)C = AC + BC 25

29 2.7 The transpose of a matrix Definition 2.3. For every matrix A R m n we can define another matrix whose rows are equal to the columns of A and whose columns are equal to the rows of A. It is called the transpose of matrix A and is denoted as A T. Note that A T R n m. Example 2.7. The transpose of A = Properties of the transpose: For all A, B R m n and all α R. (A + B) T = A T + B T (additivity) 2. (αa) T = αa T (homogeneity) 3. (A T ) T = A is AT = (AB) T = B T A T, for all A R m n, B R n t, that is the transpose of the product of two matrices is the product of the transposes in reverse order. 5. rank(a) = rank(a T ). Example We want to find the transpose of the matrix H which is a linear combination of the matrices [ ] [ ] [ ] A =, B =, C =, with coefficients α =, β = 3, γ = 2. Then H = A + 3B + 2C [ ] 0 = [ ] 3 4 = [ ] and H T = 3 6. But also A T + 3B T + 2C T = = [ 6 0 ]

30 2. A = B T = [ ] 0, B = 3 4 [ ] 2 4, A 0 T = [ ] [ ] [ ] 2 = AB = 0 3 [ ] 2 0 = (AB) T = ; 3 [ ] 2 0 = B T A T = The transpose of an upper triangular matrix is a lower triangular one. Sometimes the transpose of a square matrix is the matrix itself, i.e. A = A T. In this case the elements of A which are in a symmetric position with respect to the main diagonal are all equal, i.e. a 2 = a 2, a 3 = a 3, a 23 = a 32, etc. Definition 2.4. A square matrix A = [a ij ] R n n, is said to be symmetric if a ij = a ji, i, j =,..., n. or, equivalently, if A = A T. The set of all real symmetric matrices of order n is denoted by S n (R). Example 2.9. S = T = S is a symmetric matrix and T is not. All diagonal matrices are symmetric. 2.8 The rank of the product of two matrices Another way to look at the product of two matrices is the following. Let s consider the case in which the second factor is a column matrix, that is Ab with A R m n and b = (b,..., b n ) R n. We have a b + a 2 b a n b n a a 2 a n a 2 b + a 22 b a 2n b n Ab =. = b a 2. +b a b a 2n n., a m b + a m2 b a mn b n a m a m2 a mn so Ab is a linear combination of the columns of A with coefficients given by the components of b. Given two matrices A R m n and B R n p, we have AB = [ Ab Ab 2... Ab p ], 27

31 where b i denotes the column i of B. This means that the columns of AB are linear combinations of the columns of A. Similarly if a = (a,..., a n ) R n and B R n p we have that the product ab is a linear combination of the rows of B with coefficients given by the components of a. In the general case, if A R m n and B R n p then a B a 2 B AB =. a m B where a i denotes the row i of A. So, the rows of the product are linear combinations of the rows of B. From these observations on the row-by-column product we get the following result. Theorem 2.3. Let A R m n and B R n p then rank(ab) min(ranka, rankb). Proof Since the columns of AB are linear combinations of the columns of A we have C(AB) C(A) R m. Similarly, by observing that the rows of the product are linear combinations of the rows of B we get R(AB) R(B) R p. The consequences in terms of dimensions are and dim C(AB) dim C(A) rank(ab) ranka dim R(AB) dim R(B) In conclusion rank(ab) min(ranka, rankb). rank(ab) rankb 28

32 Chapter 3 3 Square matrices and determinant 3. The ring structure of R n n The comparison between properties of the algebraic operations defined on real numbers and the ones defined on matrices, becomes more easy and fruitful if we set in the space of square matrices. This is due to the fact that R n n is closed both under sum and product: the product (as well as the sum) of two square matrices of the same order is always doable and the result is still a square matrix of the same order. In Chapter 2 we saw that the addition between matrices has the same properties owned by addition of real numbers. What about multiplication? We already saw that, in row-by-column product, the associative property holds while the commutative one does not. It is natural to ask whether the identity element and inverse elements exist or not. The first answer is easy to give, while the second will require some work. A special (diagonal) matrix is the n n matrix with all ones on the main diagonal and all zeroes elsewhere I n = It is called the identity matrix 2 of order n, since AI n = A = I n A for all A R n n. Regarding inverse, the answer is more complicated. In R each non zero element is invertible, i.e. it has inverse with respect to multiplication. Similarly we will say that a matrix A R n n is invertible if there exists a matrix B R n n such that AB = BA = I n. It is not difficult to prove that if such a matrix exists then it is unique. So we will say that B is the inverse of A and denote it with A. For instance 2 For a rectangular matrix A R m n we have AI n = A and I ma = A. So we have a left inverse and a right inverse, but they are different. 29

33 7 A = = 7 7 [ ] [ ] 2 is the inverse of matrix A = = 7 7 = 7 [ 2 3 [ ] 7 0 = 0 7 ] [ ] 2 3 [ ] 0 0 [ ] 2 since 3 and [ ] = 7 = 7 [ ] [ ] [ ] 7 0 = 0 7 [ ] 0 0 Clearly the null matrix O n is not invertible since it is easy to prove that AO n = O n A = O n for all A R n n [. However, ] the zero matrix is not the only [ one. ] For example x y the matrix A = has no inverse. To see this, let B = be a possible 2 2 z t inverse. Then [ ] 0 = 0 [ 2 2 ] [ ] [ ] x y x + z y + t = z t 2x + 2z 2y + 2t x + z = y + t = 0 2x + 2z = 0 2y + 2t =, which is impossible. A question arises naturally: how can we tell if a square matrix is invertible? And if it is, how do we find its inverse? The following theorem answers the first question. We postpone the second problem till after introducing determinants. Theorem 3.. Let A R n n, then A is invertible if and only if A is nonsingular, i.e. ranka = n. 30

34 Proof If A is invertible, i.e. A exists, then from AA = I n and by Theorem 2.3 there follows n = ranki n = rank(aa ) min (ranka, ranka ) rank (A) hence rank(a) (and also rank(a )) is at least n. But since A R n n, the conclusion is that rank(a) = n. Conversely, if ranka = n, the columns a, a 2,... a n of A are a basis of R n. Thus every vector of R n is a linear combination of a, a 2,..., a n ; in particular so are the vectors of the standard basis. We write: namely δ ij = n e j = b j a + b 2j a b nj a n = n k= b kja k j =, 2,..., n k= b kja ik = n k= a ikb kj i, j =, 2,..., n where δ ij = if i = j and δ ij = 0 if i j. Define the matrix B = (b ij ). It is easy to check that the above set of equalities can also be written in matrix form as I n = AB. By the same argument applied to the rows instead of the columns of A, we can show the existence of matrix C such that I n = CA. Then C = CI n = CAB = I n B = B hence C = B = A. Example 3... In R 2 2 consider the matrices [ ] A = and B = 0 2 [ ]. The matrix A is of full rank, hence it is invertible, while B is not invertible since it has rank. 2. A diagonal matrix is non-singular and so invertible if and only if it has no zero elements on the main diagonal. Moreover, since in the case of diagonal matrices the product becomes entrywise, it is easy to see that if (a,..., a n ) is the main diagonal of A then also A is diagonal and its main diagonal is (a,..., a n ). The set of all invertible matrices of order n is denoted with GL n (R) and is called linear group of order n. 3

35 Properties of invertible matrices We end this section by stating some properties of invertible matrices. Let A, B GL n (R) then (A ) = A for all invertible A. This means that if A is the inverse of A, then A is the inverse of A. (AB) = B A for all invertible A and B. The inverse of a product is the product of the inverses in reverse order, since: (AB)(B A ) = A(BB )A = AI n A = AA = I n. (A T ) = ( A ) T. The transpose of the inverse is the inverse of the transpose. To show this we need to check that A T (A ) T = I n : 3.2 Definition of determinant A T (A ) T = (A A) T = I T n = I n. Let A R n n be a square matrix. The determinant of a square matrix A, a a 2... a n a 2 a a 2n denoted as det A or A =, is a real number associated a n a n2... a nn with A that is useful in finding the inverse of an invertible matrix and in solving linear systems. The integer n is called the order of the determinant. The determinant can be defined in different ways. Before we give a proper definition, we indicate how to calculate the determinant in the cases n =, 2, 3. If A is, namely A = [a ], the determinant of A is the scalar a itself: det A = det[a ] = a. [ ] a a If A is 2 2, namely A = 2, then a 2 a 22 For instance, if A = [ ] a a det 2 = a a 2 a a 22 a 2 a [ ] 2 3, then det A = 2 5 =

36 a a 2 a 3 If A is a 3 3 matrix, namely A = a 2 a 22 a 23, then a 3 a 32 a 33 a a 2 a 3 det a 2 a 22 a 23 = a a 22 a 33 + a 3 a 2 a 32 + a 2 a 23 a 3 a 3 a 32 a 33 a 3 a 22 a 3 a a 23 a 32 a 2 a 2 a 33. This rule to compute determinant is called Sarrus Rule. For instance det 0 2 = 0+0 ( )+( 2) ( 2) ( ) 0 0 = 5. 0 There exist different ways to define the determinant of a square matrix (note that it do not exists a similar definition for rectangular ones). In this notes we define the determinant by induction on the order n. We start by defining the determinant of a matrix (that is a scalar). As we saw at the beginning of the section det[a ] = a. When n >, we introduce the definition of the cofactor of the element a ij of a matrix A: it is denoted by A ij and it is ( ) i+j times the determinant of the submatrix M ij of order n (called a minor) obtained deleting the i-th row and the j-th column of A: A ij = ( ) i+j det M ij. [ ] 3 4 For instance, if A =, the cofactor of a 2 2 is A 2 = ( ) 2+ det[4] = 4; the cofactor of a is A = ( ) + det[2] = 2. Clearly ( ) i+j is always either + (if i + j is even) or (if i + j is odd); the signs alternate according to the following scheme Given A R n n, the determinant of A is defined as the sum of the elements of the first row each of them multiplied by its cofactor det (A) = a A + a 2 A a n A n = n a j A j. j= 33

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