MAC 2103 Module 10 lnner Product Spaces I 1 Learning Objectives Upon completing this module, you should be able to: 1. Define and find the inner product, norm and distance in a given inner product space. 2. Find the cosine of the angle between two vectors and determine if the vectors are orthogonal in an inner product space. 3. Find the orthogonal complement of a subspace of an inner product space. 4. Find a basis for the orthogonal complement of a subspace of ℜⁿ spanned by a set of row vectors. 5. Identify the four fundamental matrix spaces of an m x n matrix A, and know the column rank of A, the row rank of A, and the rank of A. 6. Know the equivalent statements of an n x n matrix A. http://faculty.valenciacc.edu/ashaw/ Rev.F09 Click link to download other modules. 2 1 General Vector Spaces II The major topics in this module: Inner Product Spaces, Inner Products, Norm, Distance, Fundamental Matrix Spaces, Rank, and Orthogonality http://faculty.valenciacc.edu/ashaw/ Rev.09 Click link to download other modules. 3 Definition of an Inner Product and Orthogonal Vectors A. Inner Product: Let , :V V !" "# $ % ! be any function from V ! V into ! that satisfies the following conditions for any u, v, z in V: ! ! ! ! a) !u,v" = !v,u", ! ! ! ! ! ! ! b) !u + v,z" = !u,z" + !v,z", ! ! ! ! c) !ku,v" = k!u,v", ! ! ! ! ! and ! v , v " = 0 iff v = 0. d) !v,v" # 0, ! ! Then ! u , v " defines an inner product for all u, v in V. ! ! u and v in V are Orthogonal Vectors iff !u,v" = 0. http://faculty.valenciacc.edu/ashaw/ Rev.F09 Click link to download other modules. 4 2 Definition of a Norm of a Vector B. Norm: Let ! :V " !+ = [0,#) + be any function from V into ! that satisfies the following conditions for any u, v in V: ! ! ! ! a) u ! 0, and u = 0 iff u = 0, ! ! b) su = s u , and ! ! ! ! c) u + v " u + v Triangle inequality. ! Then u defines a norm for all u in V. The special norm that is induced by an inner product is ! ! ! u = !u,u". http://faculty.valenciacc.edu/ashaw/ Rev.F09 Click link to download other modules. 5 Definition of the Distance Between Two Vectors C. Distance: Let d(!,!) :V " V # !+ = [0,$) + be any function from V ! V into ! that satisfies the following conditions for any u, v, w in V: ! ! ! ! ! ! a) d(u,v) ! 0, and d(u,v) = 0 iff u = v, ! ! ! ! b) d(u,v) = d(v,u), and ! ! ! ! ! ! c) d(u,v) " d(u,w) + d(w,v) Triangle inequality. ! ! Then d ( u , v ) defines the distance for all u, v in V. The special distance that is induced by a norm is ! ! ! ! ! ! ! ! d(u,v) = u ! v = "u ! v,u ! v#. http://faculty.valenciacc.edu/ashaw/ Rev.F09 Click link to download other modules. 6 3 How to Find an Inner Product, Norm, and Distance for Vectors in ℜⁿ? For u and v in ℜⁿ, we define ! ! ! ! u,v u v u v u v ... u v , ! " = # = 1 1 + 2 2 + + n n ! ! ! 1 ! ! u = !u,u"2 = (u #u) = u2 + u2 + ...+ u2 , 1 2 n ! ! ! ! and d(u,v) = u ! v ! ! ! ! 1 ! ! ! ! = "u ! v,u ! v#2 = (u ! v)$(u ! v) = (u ! v )2 + (u ! v )2 + ... + (u ! v )2 . 1 1 2 2 n n Note: These are our usual norm and distance in ℜⁿ. ℜⁿ is an inner product space with the dot product as its inner product. http://faculty.valenciacc.edu/ashaw/ Rev.F09 Click link to download other modules. 7 How to Find an Inner Product, Norm, and Distance for Vectors in ℜⁿ? (Cont.) Example 1: Let u = (1,2,-1) and v = (2,0,1) in ℜ³. Find the inner product, norms, and distance. ! ! ! ! !u,v" = u # v = u1v1 + u2v2 + u3v3 = (1)(2) + (2)(0) + ($1)(1) = 1, 1 ! ! ! 2 2 2 2 2 2 2 u = !u,u" = u1 + u2 + u3 = 1 + 2 + ($1) = 6, 1 ! ! ! 2 2 2 2 2 2 2 v = !v,v" = v1 + v2 + v3 = 2 + 0 + 1 = 5, and ! ! ! ! ! ! ! ! 1 d(u,v) = u $ v = !u $ v,u $ v" 2 = (1$ 2)2 + (2 $ 0)2 + ($1$ 1)2 = 9 = 3. http://faculty.valenciacc.edu/ashaw/ Rev.F09 Click link to download other modules. 8 4 How to Find an Inner Product, Norm, and Distance for Vectors (Matrices) in M22? In M22, the vectors are 2 x 2 matrices. Thus, given two matrices U and V in M22, we define T T !U,V " = tr(U V ) = tr(V U) = !V,U" = u1v1 + u2v2 + u3v3 + u4v4 . Recall: tr(A) = ! u u $ ! v v $ sum of the U = # 1 2 &,V = # 1 2 &. entries on the u u v v main diagonal of "# 3 4 %& "# 3 4 %& A. 1 2 2 2 2 2 U = !U,U" = u1 + u2 + u3 + u4 , and 1 d(U,V ) = U # V = !U # V,U # V " 2 = !U # V,U # V " 2 2 2 2 = (u1 # v1 ) + (u2 # v2 ) + (u3 # v3 ) + (u4 # v4 ) . http://faculty.valenciacc.edu/ashaw/ Rev.F09 Click link to download other modules. 9 How to Find an Inner Product, Norm, and Distance for Vectors (Matrices) in M22? (Cont.) Example 2: Let ! $ ! $ u1 u2 ! 2 '1 $ v1 v2 ! 0 9 $ U = # & = ,V = # & = u u # 3 1 & v v # 4 6 & "# 3 4 %& " % "# 3 4 %& " % be matrices in the vector space M22. Find the inner product, norms and distance for U and V in M22. T T !U,V " = tr(U V ) = tr(V U) = u1v1 + u2v2 + u3v3 + u4v4 = (2)(0) + (3)(4) + (#1)(9) + (1)(6) = 9. Recall: tr(A) = sum of the entries on the main diagonal of A. http://faculty.valenciacc.edu/ashaw/ Rev.F09 Click link to download other modules. 10 5 How to Find an Inner Product, Norm, and Distance for Vectors (Matrices) in M22? (Cont.) 1 2 2 2 2 2 2 2 2 2 U = !U,U" = u1 + u2 + u3 + u4 = 2 + (#1) + 3 + 1 = 15, 1 2 2 2 2 2 2 2 2 2 V = !V,V " = u1 + u2 + u3 + u4 = 0 + 9 + 4 + 6 = 133, and 1 d(U,V ) = U # V = !U # V,U # V " 2 = !U # V,U # V " 2 2 2 2 = (u1 # v1 ) + (u2 # v2 ) + (u3 # v3 ) + (u4 # v4 ) = (2 # 0)2 + ((#1) # 9)2 + (3 # 4)2 + (1# 6)2 = 4 + 100 + 1+ 25 = 130. http://faculty.valenciacc.edu/ashaw/ Rev.F09 Click link to download other modules. 11 How to Find an Inner Product, Norm, and Distance for Vectors (Polynomials) in P2? Let P be the set of all polynomials of degree less than or equal 2 ! ! to two. For two polynomials p,q !P2 , 0 1 2 p(x) = a0 x + a1x + a2 x , 0 1 2 q(x) = b0 x + b1x + b2 x ! ! we define !p,q" = a0b0 + a1b1 + a2b2 , 1 ! ! ! 2 2 2 2 p = !p, p" = a0 + a1 + a2 , and ! ! ! ! ! ! ! ! 1 d(p,q) = p # q = !p # q, p # q" 2 2 2 2 = (a0 # b0 ) + (a1 # b1 )1 + (a3 # b3 ) . http://faculty.valenciacc.edu/ashaw/ Rev.F09 Click link to download other modules. 12 6 How to Find an Inner Product, Norm, and Distance for Vectors (Polynomials) in P2? (Cont.) Example 3: Find the inner product, norms, and distance for p and q in P2, where p(x) = 8 ! 3x + 5x2 and q(x) = 1! 2x + 7x2 . ! ! !p,q" = a0b0 + a1b1 + a2b2 = (8)(1) + (#3)(#2) + (5)(7) = 8 + 6 + 35 = 49, 1 ! ! ! 2 2 2 2 2 2 2 p = !p, p" = a0 + a1 + a2 = 8 + (#3) + 5 = 98 = 7 2, 1 ! ! ! 2 2 2 2 2 2 2 q = !q,q" = b0 + b1 + b2 = 1 + (#2) + 7 = 54 = 3 6, and 1 ! ! ! ! ! ! ! ! 2 2 2 2 d(p,q) = p # q = !p # q, p # q" = (a0 # b0 ) + (a1 # b1 )1 + (a3 # b3 ) = (8 # 1)2 + ((#3) # (#2))2 + (5 # 7)2 = 49 + 1+ 4 = 54 = 3 6. http://faculty.valenciacc.edu/ashaw/ Rev.F09 Click link to download other modules. 13 How to Find an Inner Product, Norm, and Distance for Vectors (Functions) in C[a,b]? Let C[a,b] be the set of all continuous functions on [a,b]. For all f and g in C[a,b], we define b ! ! ! f ,g" = # f (x)g(x)dx, a 1 b 2 b ! ! ! 1 $ ' 2 2 2 f = ! f , f " = & #[ f (x)] dx) = # f (x)dx, % a ( a 1 b 2 ! ! ! ! 1 $ ' ! ! ! ! 2 2 and d( f ,g) = f * g = ! f * g, f * g" = & #[ f (x) * g(x)] dx) . % a ( http://faculty.valenciacc.edu/ashaw/ Rev.F09 Click link to download other modules. 14 7 How to Find an Inner Product, Norm, and Distance for Vectors (Functions) in C[a,b]? (Cont.) Example 4: Let f = f(x) = 1 and g = g(x) = -x. Find the inner product, norms, and distance on C[2,4]. 4 4 4 ! ! x2 ! f ,g" = # f (x)g(x)dx = # (1)($x)dx = $ = ($8) $ ($2) = $6, 2 2 2 2 1 4 2 4 4 ! ! ! 1 % ( 4 f = ! f , f " 2 = [ f (x)]2 dx = f 2 (x)dx = dx == x = 2, ' # * # # 2 & 2 ) 2 2 1 4 2 4 4 3 4 1 % ( x 14 ! ! ! 2 2 2 2 g = !g,g" = ' #[g(x)] dx* = # g (x)dx = # ($x) dx == = 2 , & 2 ) 2 2 3 2 3 1 4 2 ! ! ! ! 1 % ( ! ! ! ! 2 2 and d( f ,g) = f $ g = ! f $ g, f $ g" = ' #[ f (x) $ g(x)] dx* & 2 ) 4 4 4 x3 2 = # (1+ x)2 dx = # (1+ 2x + x2 )dx = (x + x2 + ) = 7 .