Chapter 2 General Vector Spaces Outline : Real vector spaces Subspaces Linear independence Basis and dimension Row Space, Column Space, and Nullspace 2 Real Vector Spaces 2 Example () Let u and v be vectors in R n and k a scalar ( a real number), then we can define Addition: u+v, Scalar multiplication: ku, kv (2) Let P 2 denote the set of all polynomials of the form p=ax 2 +bx+c For example, p=, p=x, p=x+5 are in P 2 Given any two polynomials p= ax 2 +bx+c and q= a'x 2 +b'x+c', and a scalar k we can also define Addition: p+q= (a+a')x 2 +(b+b')x+(c+c'), Scalar multiplication: kp= kax 2 +kbx+kc (3) Let M 22 be the set of all 2 2 matrices Then we can also define the addition and scalar multiplication in the ordinary way
About the turn of last century, mathematicians noticed that, in the development of both mathematics and its applications, many sets of objects were turning up quite naturally with properties of addition and scalar multiplication in accordance with (A) through (A7) listed bellow Moreover, these properties proved so useful that it was worth while giving a generic name to all collections of objects that possess such two operations rules Nowadays they are called vector spaces( or linear spaces ) 22 Definition Let V be an arbitrary nonempty set of objects on which two operations are defined, addition and multiplication by scalars (numbers) By addition, we mean a rule for associating with each pair of objects u and v in V, an object called the sum of u and v; by scalar multiplication, we mean a rule for associating with each scalar k and each object u in V an object k u, called the scalar multiple of u by k If the following axioms are satisfied by all objects u, v, w in V and all scalars k, l, then we call V a vector space and we call the objects in V vectors: (A) If u and v are objects in V, then u + v V (A2) u + v = v + u (A3) u + ( v + w) = ( u + v) + w (A4) There is an object in V, called the zero vector for V such that + u = u + = u for all u in V (A5) For each u in V, there is an object u in V, called the negative of u, such that (A6) If k R, u V, then ku V (A7) k( u + v) = ku + kv (A8) u u+(-u)=(-u)+u= ( k + l) u = ku + lu, k( lu) = ( kl) u = u
Vector space R 2, R 3,, R n, P 2, P 3,,P n,, M 2 2, M 2 3,, Exercise: Verify the conditions (A3) and (A7) for u=(,-), v=(,2), w=(3,3), k=2, l=5 23 Proposition Let V be a vector space and let u V, k R Then (a) u = (b) k = (c) ( ) u = u (d) If ku =, then k = or u = 24 Example n The set V = R with the standard operations of addition and scalar multiplication is a vector space 2 The set V of all 2 2 matrices with real entries with the standard operations of matrix addition and matrix scalar multiplication It is denoted by M 22 3 The set V of all m n matrices with real entries under the standard operations of matrix addition and matrix scalar multiplication Denote by M mn 4 Let V be the set of real-valued functions defined on the entire real line (, ) Let f = f ( x), g = g( x) be two such functions in V and let k R Then
( f + g)( x) = f ( x) + g( x ), ( kf )( x) = kf( x) Then V is a vector space under the defined operations of addition and scalar multiplication Denote by F(, ) 5 The set P n of all polynomials of degree n is a vector space under the ordinary addition and scalar multiplication of polynomials 6 The set V={ (x,y,z): 2x+4y-z=}, under the standard operations of addition and scalar multiplication for vectors in R 3 is a vector space 7 Let V = { } and define addition and scalar multiplication as follows: + = ; k = This called the trivial space 25 Example The set V = R 2 where addition and scalar multiplication are defined as follows: For u = ( u, u2), v = ( v, v2 ), define u + v = ( u + v, u + v ); ku = ( ku, ) 2 2 () Is condition (A4) satisfied? (2) Is condition (A7) satisfied? (3) Is condition (A8) satisfied? Exercise 2 Let V=R* be the set of all positive real numbers For x,y in V and any scalar k, define x+y=xy, kx=x k Determine if V is a vector space 2 Prove: in a vector space V if k is a scalar and u is a vector such that ku= then k= or u= 3 Show that in any vector space the zero vector is unique 4* Let V=R 2 Suppose the addition of vectors is the usual addition and the scalar multiplication is defined by k(x,y)=(,) if k=, k(x,y)=(kx, y/k) Determine whether the axioms (A7) is satisfied
5* Find a set V and define the addition + and scalar multiplication such that (V,+,) satisfies all axioms except (A7) 22 Subspaces From a given vector space we can construct other new vector spaces One important type of spaces obtained from V are the subspaces of V 22 Definition A nonempty subset W of a vector space V is called a subspace of V if W is itself a vector space under the addition and scalar multiplication defined on V 222 Remarks () It can be shown that a nonempty subset W of a vector space V is a subspace if and only if it is closed under addition and scalar multiplication, that is u, v W implies u+v W and ku W, kv W for any scalar k V u v u+v ku kv W (2) If B is a subspace and v, v2, K, v r are from B then k v +k 2 v 2 + +k r v r is also in B for any scalars k, k 2,,k r
223 Example () Let D nn be the set of all the n n diagonal matrices Then D nn is a subspace of M nn (2) W={(x,y): x+2y=} is a subspace of R 2 Is U={(x,y): x+y-=} a subspace of R 2? (3) Let B={ (,2,,), (,,,-)} Consider W={ k(,2,,)+l(,,,-): k,l R} Then W is a subspace, which is called the subspace spanned by B (4) P 2 is a subspace of P 3 Generally, P m is a subspace of P n if m<n (5) Every V has at least two trivial subspaces, V itself and the zero space {} (6) If Ax= is a homogeneous linear system of m equations in n unknowns, then the set of solution vectors is a subspace in R n Let e=(,), e2=(,) Then (2,3)=(2,)+(,3)=2(,)+3(,)=2 e+3 e2 We say the vector (2,3) is a linear combination of e and e 2 Every vector in R 2 is a linear combination of e and e 2 224 Definition A vector w is called a linear combination of the vectors v, v2, K, v r, if w = k v + k 2 v 2 + K + k r v r, for some scalars k, k, K, 2 k r R 225 Definition Let S = { v, v2, K, vr} be a set of vectors in a vector space V Let W be the subspace of V consisting of all linear combinations of the vectors in S Then W is called the space spanned by v, v2, K, v r, and we say that the vectors v, v2, K, v r span W We write W = span ( S) or W = span { v, v, K 2, vr} Exercise Show that span{ (,2),(2,-)}=R 2 226 Remarks ()W = span { v, v2, K, vr} is the smallest subspace of V that contains v, v2, K, v r in the sense that every subspace of V that contains v, v, K, v r must contain W 2
(2) If S = { v, v, K 2, vr} and S = { w, w 2, K, w k} are two sets of vectors in a vector space V, then Span(S)=Span(S') if and only if each vector in S is a linear combination of those in S, and conversely, each vector in S is a linear combination of those in S Exercise 22 Determine which of the following are subspaces of the vector space M nn (a) The set of n n symmetric matrices( A is symmetric if A T =A) (b) The set of n n lower triangular matrices (c) The set of all non-invertible n n matrices 2 Determine which of the following are subspaces of F(, ): (a) The set of continuous functions (b) The set of differentiable functions (c) The set of all increasing functions ( f is increasing if x< y implies f(x)<f(y)) 3 Consider the vectors u = (,2 ) and v = ( 6, 4,2) in R 2 Show that w=(5,2,3) is a linear combination of u and v, and that w = ( 4,, 8 ) is not a linear combination of u and v 4 Determine whether v = (,,2), v = (,, ), v = ( 2, 3, ) span the vector space R 2 2 3 5 Prove: if a vector u is a linear combination of v, v 2, and each v i is a linear combination of w and w 2, then u is a linear combination of w and w 2 Extend this result 23 Linear Independence For a given space V, generally there are more than one subset that span V However, among all such spanning subsets there exist minimal one that contains the smallest number of vectors Such a minimal spanning set is called a basis of the space V
Consider the two groups of vectors B= {(,2), (,),(,)} and C={(,2),(2,3)} We can find scalars k,l,m, not all zero, such that k(,2)+l(,,)+m(,)=(,) But if k(,2)+l(2,3)=(,) then the only possibility is k=,l= 23 Definition Let S = { v, v, K, v } be a nonempty set of vectors Then the vector equation 2 r k v + k 2 v 2 + K k r v r = has at least one solution, namely k =, k 2 =, K, k r = If this is the only solution, then S is called a linearly independent set If there are other solutions, then S is called a linearly dependent set 232 Examples () { (,), (,)} is linearly independent in R 2 In general, {(,,,), (,,,,), (,,,)} is linearly independent in R n (2) {,x,x 2,, x n } is linearly independent in P n (3) {,,, } is linearly independent in M 22 233 Proposition A set S with two or more vectors is (a) Linearly dependent if and only if at least one of the vectors in S is expressible as a linear combination of the other vectors in S (b) Linearly independent if and only if no vector of S is expressible as a linear combination of the other vectors in S
234 Examples () If v = ( 2,,, 3), v = (,2, 5, ), v = ( 7,, 5, 8), show that 2 3 S = { v, v, v } 2 3 is linearly dependent (2) Determine whether v = (, 2, 3), v = ( 5, 6, ), v = ( 3,2, ) form a linearly dependent set in R 2 2 3 (3) Show that {+x, x+x 2 } is linearly independent in P 2 (4) Show that the functions f = x, f = sin x form a linearly independent set of vectors in the 2 space F(, ) of functions on the real line Exercise 23 Show that the polynomials 2 2 p = x, p = 5+ 3x 2x, p = + 3x x form a linearly dependent set in P 2 2 3 2 Prove: (a) A finite set of vectors that contains the zero vector is linearly dependent (b) A set with exactly two vectors is linearly independent if and only if neither vector is a scalar multiple of the other 3 Let S = { v, v, K 2, vr} be a set of vectors in R2 If r > n, then S is linearly dependent 4 Prove if B is a subset of a linearly independent set S, then B is also linearly independent Is a subset of linearly dependent set always linearly dependent? 5 Let S and B be two linearly independent sets of vectors in V Show that Span{S} Span{B } ={} iff B S is linearly independent Find two linearly independent sets such that their union is linearly dependent 6 Prove: in the space C(a,b) of all continuous functions defined on [a,b], the set {, x, cos
x} is linearly independent Is {, sin 2 x, cos 2 x} linearly independent? 24 Basis and Dimension 24 Definition Let V be a vector space and let S = { v, v2, K, vr} be a set of vectors in V Then S is called a basis for V if the following two conditions hold: S is linearly independent; S spans V 242 Theorem Let S = { v, v2, K, vn} be a basis for a vector space V Then every vector v can be expressed as v = c v + c 2 v 2 + K + c n v n in a unique way 243 Definition The scalars c, c 2, K, c n in Theorem 42 are called the coordinates of v relative to the basis S The vector ( c, c 2, K, c n ), denoted by [v] S, in R n is called the coordinate vector of v relative to S 244 Remarks For a given basis, the function [] S : V R n is a bijection 245 Examples () Let e = (,,, K, ), e = (,,,, ),, e = (,,,, 2 K K n K ) Then S = { e, e, K 2, e n} is called the standard basis for R n (2) {,x,x 2,, x n } is called the standard basis of P n (3) There is no a finite basis for F(, ) 246 Theorem All bases for a vector spaces have the same number of vectors 247 Definition () A nonzero vector space is called finite-dimensional if it contains a set of vectors { v, v2, K, v n } that forms a basis If no such set exists, V is called infinite dimensional In addition, the zero vector space is considered to be finite -dimensional,
(2) The dimension of a finite -dimensional vector space V, denoted by dim(v) is defined to be the number of vectors in a basis for V In addition, we define the zero vector space to have dimension zero 247 Theorem* If V is an n-dimensional vector space, then: (a) Every set with more than n vectors is linearly dependent (b) If S has n vectors and spans V, then S is a basis (c) If S is linearly independent and has n vectors, then S is a basis 248 Theorem * Let S be a finite set of vectors in a finite-dimensional vector space V (a) If S spans V, but is not a basis for V, then S can be reduced to a basis for V by removing appropriate vectors from S (b) If S is a linearly independent set that is not a basis for V, then S can be enlarged to a basis for V by inserting appropriate vectors into S 249 Examples () dim(r n )=n S = { e, e, K 2, e n} is a basis for Rn (2) Let v = (,2, ), v = ( 2,9, ), v = ( 3, 3, 4) Show that the set 2 3 S = { v, v2, v3 } is a basis for R3 (3) Let S = { v, v2, v3 } be the basis for R3 in (2) (a) Find the co-ordinate vector of v = ( 5,,9) with respect to S (b) Find the vector v in R 3 whose co-ordinate vector with respect to S is ( v) S = ( 3,,2) (4) S={,x,, x n } is a basis for the vector space P n, so dim( P n )=n 2 The coordinate vector of the polynomial p = a + a x + a x relative to the basis 2 S = {, x, x } for P 2 is [p] S =(a,a,a 2 ) 2
(5) Let P[x] be the vector space of all polynomials in x Then B={,x,x 2,, x n, } is a basis of P[x] So the dimension of P[x] is infinite (5) Determine a basis for and the dimension of the solution space of the homogeneous system 2x + 2x x + x = 2 3 5 x x + 2x 3x + x = 2 3 4 5 x + x 2x x = 2 3 5 x + x + x = 3 4 5 Exercise 24 () Prove that S={ (,2,),(,,),(,,)} is a basis of R 3 Find the coordinate vector of v=(,2,5) with respect to S (2) Suppose the coordinate vector of a vector u with respect to S is (2,-,4) Determine the vector u 2 Let S = { v, v2, K, vr} be a linearly independent set in the vector space V Then S is a basis for the subspace span{s} 3 Prove S={ +x, x+2x 2, + x 2 } is a basis of P 2 Find [a x 2 +bx+c] S 4 Find the dimension of the solution space of the following linear system 2x-y+z -w= x +2z = 2y +w= 5 Prove if S = { v, v, K 2, vr} is a basis for V, then for any vector v in V, { v,v,v,,v } is 2 r linearly dependent 6 Prove that for any vectors u,v and w in a vector space V, { u-v,v-w, w-u} is linearly dependent
7 Let { v,v 2, v 3 } be a basis of a space V Prove: (i) { v 2, v, v 3 } is a basis (ii) { kv,v 2, v 3 } is a basis for any nonzero scalar k (iii) { v,kv +v 2, v 3 } is a basis 8 Show that if { v,v 2,,v r } is a basis of V, then { v, v +v 2,, v +v 2 + +v r } is also a basis of V 25 Row Space, Column Space, and Nullspace Given an m n matrix A The row vectors of A are vectors in R n and the column vectors of A are vectors in R m, and the solution vectors of the linear system Ax= are vectors in R n From these vectors we can construct three vector spaces A Row space column space nullspace I I I R n R m R n 25 Definitions Let A be an m n matrix Then the subspace of R n spanned by the row vectors of A is called the row space of A, and the subspace of R m spanned by the column vectors of A is called the column space of A The solution space of the homogeneous system of equations Ax =, which is a subspace of R n, is called the nullspace of A
252 Proposition () Elementary row operations do not change the nullspace of a matrix (2) Elementary row operations do not change the row space of a matrix (3) If a matrix R is in row-echelon form, then the nonzero row vectors form a basis for the row space of R Recall that two matrices are said to be row equivalent if one of them can be obtained from another by using elementary row operations A B 253 Proposition Let A and B be row equivalent matrices Then (a) A given set of column vectors of A is linearly independent if and only if the corresponding column vectors of B are linearly independent (b) A given set of column vectors of A forms a basis for the column space of A if and only if the corresponding column vectors of B form a basis for the column space of B (c) If a matrix A is in row-echelon form, then the column vectors that contains a leading form a basis for the column space of A 254 Remarks () By 52, to find a basis for the row space of A, we first reduce A to its row echelon form Then the rows containing leading form a basis of the row space (2) To find a basis of the column space of A, just choose those columns in A corresponding to the columns in the row echelon form of A that contains a leading (3) From () and (2) it follows that the row space and the column space of a matrix A have the same dimension(why?)
255 Example Let S={ (,,), (,2,3), (,,3)} Find a basis of the space Span{S} Solution: Let A= 2 3 Then Span{S} is the column space of A 3 Reduce A to row echelon form A'= The first and second columns of A' contain leading, so the corresponding columns in A form a basis of A, and therefore of Span{S}, that is { (,,),(,2,3)} 256 Algorithm: Given a set of vectors S = { v, v2, K, vk} in R n To find a subset of these vectors that form a basis for span(s): Step : Form the matrix A having v, v2, K, v k as its column vectors Step 2: Reduce the matrix A to its reduced row-echelon form R, and let w, w2, K, w k be the column vectors of R Step 3: Identify the column vectors that contain a leading These vectors form a basis for span(s) The corresponding column vectors of A are the basis vectors for span( S) 257 Definition Let A be a matrix The dimension of the row space ( or the column space) of A is called the rank of A, denoted by rank( A) The dimension of the nullspace of A is called the nullity of A, denoted by nullity(a) 258 Theorem( Dimension theorem for matrices) Let A have n columns Then rank(a) + nullity(a) = n
259 Theorem* n n Given A n n with corresponding linear transformation T:R R, where T(u)=Au Then the following statements are equivalent (a) A is invertible (b) Ax = has only the trivial solution (c) The reduced row-echelon form of A is I n (d) A is a product of elementary matrices (e) Ax (f) Ax (g) det(a) = b is consistent for every n matrix b = b has exactly one solution (h) The range of T is R n (i) T is injective (j) The column vectors of A are linearly independent (k) The row vectors of A are linearly independent (l) The column vectors of A span R n (m) The row vectors of A span R n (n) The column vectors of A form a basis for R n (o) The row vectors of A form a basis for R n (p) Rank(A) = n (q) Nullity(A) =
Exercise 5 Find a basis for the space spanned by the vectors v = (, 2,,, 3), v = ( 2, 5, 3, 2, 6), v = (, 55,,, ), v = ( 2, 6, 8, 8, 6) 2 3 4 What is the dimension of the spanned subspace? 2 Find the rank and the nullity of the matrices below: 2 A= 2, B= 2 Verify the result that rank(a) + nullity(a) = n 3 Let A be a 7 4 matrix Show that the row vectors of A must be linearly dependent 4Give an example to show that elementary row operations may change the column space of a matrix 5 Prove that a system of linear equation Ax = b is consistent if and only if b is in the column space of A 6 Explain why the row space and the column space of a matrix have the same dimension 7* Let A and B be two n n matrices Determine which of the following is true for all A and B: (a) rank(ab)=rank(a)rank(b), (b) rank(ab)=max{ rank(a),rank(b)}, (c) rank(ab)=min{rank(a),rank(b)}