Duality, part 1: Dual Bases and Dual Maps Notation F denotes either R or C. V and W denote vector spaces over F. Define ': R3 ! R by '(x; y; z) = 4x − 5y + 2z. Then ' is a linear functional on R3. n n Fix (b1;:::; bn) 2 C . Define ': C ! C by '(z1;:::; zn) = b1z1 + ··· + bnzn: Then ' is a linear functional on Cn. Define ': P(R) ! R by '(p) = 3p00(5) + 7p(4). Then ' is a linear functional on P(R). R 1 Define ': P(R) ! R by '(p) = 0 p(x) dx. Then ' is a linear functional on P(R). Examples: Linear Functionals Definition: linear functional A linear functional on V is a linear map from V to F. In other words, a linear functional is an element of L(V; F). n n Fix (b1;:::; bn) 2 C . Define ': C ! C by '(z1;:::; zn) = b1z1 + ··· + bnzn: Then ' is a linear functional on Cn. Define ': P(R) ! R by '(p) = 3p00(5) + 7p(4). Then ' is a linear functional on P(R). R 1 Define ': P(R) ! R by '(p) = 0 p(x) dx. Then ' is a linear functional on P(R). Linear Functionals Definition: linear functional A linear functional on V is a linear map from V to F. In other words, a linear functional is an element of L(V; F). Examples: Define ': R3 ! R by '(x; y; z) = 4x − 5y + 2z. Then ' is a linear functional on R3. Define ': P(R) ! R by '(p) = 3p00(5) + 7p(4). Then ' is a linear functional on P(R). R 1 Define ': P(R) ! R by '(p) = 0 p(x) dx. Then ' is a linear functional on P(R). Linear Functionals Definition: linear functional A linear functional on V is a linear map from V to F. In other words, a linear functional is an element of L(V; F). Examples: Define ': R3 ! R by '(x; y; z) = 4x − 5y + 2z. Then ' is a linear functional on R3. n n Fix (b1;:::; bn) 2 C . Define ': C ! C by '(z1;:::; zn) = b1z1 + ··· + bnzn: Then ' is a linear functional on Cn. R 1 Define ': P(R) ! R by '(p) = 0 p(x) dx. Then ' is a linear functional on P(R). Linear Functionals Definition: linear functional A linear functional on V is a linear map from V to F. In other words, a linear functional is an element of L(V; F). Examples: Define ': R3 ! R by '(x; y; z) = 4x − 5y + 2z. Then ' is a linear functional on R3. n n Fix (b1;:::; bn) 2 C . Define ': C ! C by '(z1;:::; zn) = b1z1 + ··· + bnzn: Then ' is a linear functional on Cn. Define ': P(R) ! R by '(p) = 3p00(5) + 7p(4). Then ' is a linear functional on P(R). Linear Functionals Definition: linear functional A linear functional on V is a linear map from V to F. In other words, a linear functional is an element of L(V; F). Examples: Define ': R3 ! R by '(x; y; z) = 4x − 5y + 2z. Then ' is a linear functional on R3. n n Fix (b1;:::; bn) 2 C . Define ': C ! C by '(z1;:::; zn) = b1z1 + ··· + bnzn: Then ' is a linear functional on Cn. Define ': P(R) ! R by '(p) = 3p00(5) + 7p(4). Then ' is a linear functional on P(R). R 1 Define ': P(R) ! R by '(p) = 0 p(x) dx. Then ' is a linear functional on P(R). = (dim V)(dim F) = dim V dim V0 = dim V Suppose V is finite-dimensional. Then V0 is also finite-dimensional and dim V0 = dim V. Proof dim V0 = dim L(V; F) The Dual Space and Its Dimension Definition: dual space, V0 The dual space of V, denoted V0, is the vector space of all linear functionals on V. In other words, V0 = L(V; F). = (dim V)(dim F) = dim V Proof dim V0 = dim L(V; F) The Dual Space and Its Dimension Definition: dual space, V0 The dual space of V, denoted V0, is the vector space of all linear functionals on V. In other words, V0 = L(V; F). dim V0 = dim V Suppose V is finite-dimensional. Then V0 is also finite-dimensional and dim V0 = dim V. = (dim V)(dim F) = dim V The Dual Space and Its Dimension Definition: dual space, V0 The dual space of V, denoted V0, is the vector space of all linear functionals on V. In other words, V0 = L(V; F). dim V0 = dim V Suppose V is finite-dimensional. Then V0 is also finite-dimensional and dim V0 = dim V. Proof dim V0 = dim L(V; F) = dim V The Dual Space and Its Dimension Definition: dual space, V0 The dual space of V, denoted V0, is the vector space of all linear functionals on V. In other words, V0 = L(V; F). dim V0 = dim V Suppose V is finite-dimensional. Then V0 is also finite-dimensional and dim V0 = dim V. Proof dim V0 = dim L(V; F) = (dim V)(dim F) The Dual Space and Its Dimension Definition: dual space, V0 The dual space of V, denoted V0, is the vector space of all linear functionals on V. In other words, V0 = L(V; F). dim V0 = dim V Suppose V is finite-dimensional. Then V0 is also finite-dimensional and dim V0 = dim V. Proof dim V0 = dim L(V; F) = (dim V)(dim F) = dim V Example: Consider the standard basis e1;:::; en of Fn. For 1 ≤ j ≤ n, define ' by Dual basis is a basis of j the dual space 'j(x1;:::; xn) = xj for (x ;:::; x ) 2 Fn. Clearly ( Suppose dim V < 1. Then 1 n 1 if k = j; 'j(ek) = the dual basis of a basis of V 6= : 0 if k j is a basis of V0. Thus '1;:::;'n is the dual basis of the n standard basis e1;:::; en of F . The Dual Basis Definition: dual basis If v1;:::; vn is a basis of V, then the dual basis of v1;:::; vn is the list 0 '1;:::;'n of elements of V , where each 'j is the linear functional on such that V ( 1 if k = j; 'j(vk) = 0 if k 6= j: For 1 ≤ j ≤ n, define ' by Dual basis is a basis of j the dual space 'j(x1;:::; xn) = xj for (x ;:::; x ) 2 Fn. Clearly ( Suppose dim V < 1. Then 1 n 1 if k = j; 'j(ek) = the dual basis of a basis of V 6= : 0 if k j is a basis of V0. Thus '1;:::;'n is the dual basis of the n standard basis e1;:::; en of F . The Dual Basis Definition: dual basis If v1;:::; vn is a basis of V, then the dual basis of v1;:::; vn is the list 0 '1;:::;'n of elements of V , where each 'j is the linear functional on such that V ( 1 if k = j; 'j(vk) = 0 if k 6= j: Example: Consider the standard basis e1;:::; en of Fn. Dual basis is a basis of the dual space Clearly ( Suppose dim V < 1. Then 1 if k = j; 'j(ek) = the dual basis of a basis of V 6= : 0 if k j is a basis of V0. Thus '1;:::;'n is the dual basis of the n standard basis e1;:::; en of F . The Dual Basis Definition: dual basis If v1;:::; vn is a basis of V, then the dual basis of v1;:::; vn is the list 0 '1;:::;'n of elements of V , where each 'j is the linear functional on such that V ( 1 if k = j; 'j(vk) = 0 if k 6= j: Example: Consider the standard basis e1;:::; en n of F . For 1 ≤ j ≤ n, define 'j by 'j(x1;:::; xn) = xj n for (x1;:::; xn) 2 F . Dual basis is a basis of the dual space Suppose dim V < 1. Then the dual basis of a basis of V is a basis of V0. The Dual Basis Definition: dual basis If v1;:::; vn is a basis of V, then the dual basis of v1;:::; vn is the list 0 '1;:::;'n of elements of V , where each 'j is the linear functional on such that V ( 1 if k = j; 'j(vk) = 0 if k 6= j: Example: Consider the standard basis e1;:::; en n of F . For 1 ≤ j ≤ n, define 'j by 'j(x1;:::; xn) = xj for (x ;:::; x ) 2 Fn. Clearly ( 1 n 1 if k = j; 'j(ek) = 0 if k 6= j: Thus '1;:::;'n is the dual basis of the n standard basis e1;:::; en of F . The Dual Basis Definition: dual basis If v1;:::; vn is a basis of V, then the dual basis of v1;:::; vn is the list 0 '1;:::;'n of elements of V , where each 'j is the linear functional on such that V ( 1 if k = j; 'j(vk) = 0 if k 6= j: Example: Consider the standard basis e1;:::; en of Fn. For 1 ≤ j ≤ n, define ' by Dual basis is a basis of j the dual space 'j(x1;:::; xn) = xj for (x ;:::; x ) 2 Fn. Clearly ( Suppose dim V < 1. Then 1 n 1 if k = j; 'j(ek) = the dual basis of a basis of V 6= : 0 if k j is a basis of V0. Thus '1;:::;'n is the dual basis of the n standard basis e1;:::; en of F . Example: Define D: P(R) !P(R) by Dp = p0. Suppose ' is the linear functional on P(R) defined by '(p) = p(3).