Duality, part 1: Dual Bases and Dual Maps Notation

F denotes either R or C.

V and W denote vector spaces over F. Define ϕ: R3 → R by ϕ(x, y, z) = 4x − 5y + 2z. Then ϕ is a linear functional on R3. n n Fix (b1,..., bn) ∈ C . Define ϕ: C → C by ϕ(z1,..., zn) = b1z1 + ··· + bnzn. Then ϕ is a linear functional on Cn. Define ϕ: P(R) → R by ϕ(p) = 3p00(5) + 7p(4). Then ϕ is a linear functional on P(R). R 1 Define ϕ: P(R) → R by ϕ(p) = 0 p(x) dx. Then ϕ 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) ∈ C . Define ϕ: C → C by ϕ(z1,..., zn) = b1z1 + ··· + bnzn. Then ϕ is a linear functional on Cn. Define ϕ: P(R) → R by ϕ(p) = 3p00(5) + 7p(4). Then ϕ is a linear functional on P(R). R 1 Define ϕ: P(R) → R by ϕ(p) = 0 p(x) dx. Then ϕ 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 ϕ: R3 → R by ϕ(x, y, z) = 4x − 5y + 2z. Then ϕ is a linear functional on R3. Define ϕ: P(R) → R by ϕ(p) = 3p00(5) + 7p(4). Then ϕ is a linear functional on P(R). R 1 Define ϕ: P(R) → R by ϕ(p) = 0 p(x) dx. Then ϕ 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 ϕ: R3 → R by ϕ(x, y, z) = 4x − 5y + 2z. Then ϕ is a linear functional on R3. n n Fix (b1,..., bn) ∈ C . Define ϕ: C → C by ϕ(z1,..., zn) = b1z1 + ··· + bnzn. Then ϕ is a linear functional on Cn. R 1 Define ϕ: P(R) → R by ϕ(p) = 0 p(x) dx. Then ϕ 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 ϕ: R3 → R by ϕ(x, y, z) = 4x − 5y + 2z. Then ϕ is a linear functional on R3. n n Fix (b1,..., bn) ∈ C . Define ϕ: C → C by ϕ(z1,..., zn) = b1z1 + ··· + bnzn. Then ϕ is a linear functional on Cn. Define ϕ: P(R) → R by ϕ(p) = 3p00(5) + 7p(4). Then ϕ 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 ϕ: R3 → R by ϕ(x, y, z) = 4x − 5y + 2z. Then ϕ is a linear functional on R3. n n Fix (b1,..., bn) ∈ C . Define ϕ: C → C by ϕ(z1,..., zn) = b1z1 + ··· + bnzn. Then ϕ is a linear functional on Cn. Define ϕ: P(R) → R by ϕ(p) = 3p00(5) + 7p(4). Then ϕ is a linear functional on P(R). R 1 Define ϕ: P(R) → R by ϕ(p) = 0 p(x) dx. Then ϕ 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 ϕ by Dual basis is a basis of j the dual space ϕj(x1,..., xn) = xj for (x ,..., x ) ∈ Fn. Clearly ( Suppose dim V < ∞. Then 1 n 1 if k = j, ϕj(ek) = the dual basis of a basis of V 6= . 0 if k j is a basis of V0. Thus ϕ1, . . . , ϕn 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 ϕ1, . . . , ϕn of elements of V , where each ϕj is the linear functional on such that V ( 1 if k = j, ϕj(vk) = 0 if k 6= j. For 1 ≤ j ≤ n, define ϕ by Dual basis is a basis of j the dual space ϕj(x1,..., xn) = xj for (x ,..., x ) ∈ Fn. Clearly ( Suppose dim V < ∞. Then 1 n 1 if k = j, ϕj(ek) = the dual basis of a basis of V 6= . 0 if k j is a basis of V0. Thus ϕ1, . . . , ϕn 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 ϕ1, . . . , ϕn of elements of V , where each ϕj is the linear functional on such that V ( 1 if k = j, ϕj(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 < ∞. Then 1 if k = j, ϕj(ek) = the dual basis of a basis of V 6= . 0 if k j is a basis of V0. Thus ϕ1, . . . , ϕn 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 ϕ1, . . . , ϕn of elements of V , where each ϕj is the linear functional on such that V ( 1 if k = j, ϕj(vk) = 0 if k 6= j.

Example: Consider the standard basis e1,..., en n of F . For 1 ≤ j ≤ n, define ϕj by ϕj(x1,..., xn) = xj n for (x1,..., xn) ∈ F . Dual basis is a basis of the dual space

Suppose dim V < ∞. 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 ϕ1, . . . , ϕn of elements of V , where each ϕj is the linear functional on such that V ( 1 if k = j, ϕj(vk) = 0 if k 6= j.

Example: Consider the standard basis e1,..., en n of F . For 1 ≤ j ≤ n, define ϕj by ϕj(x1,..., xn) = xj for (x ,..., x ) ∈ Fn. Clearly ( 1 n 1 if k = j, ϕj(ek) = 0 if k 6= j. Thus ϕ1, . . . , ϕn 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 ϕ1, . . . , ϕn of elements of V , where each ϕj is the linear functional on such that V ( 1 if k = j, ϕj(vk) = 0 if k 6= j.

Example: Consider the standard basis e1,..., en of Fn. For 1 ≤ j ≤ n, define ϕ by Dual basis is a basis of j the dual space ϕj(x1,..., xn) = xj for (x ,..., x ) ∈ Fn. Clearly ( Suppose dim V < ∞. Then 1 n 1 if k = j, ϕj(ek) = the dual basis of a basis of V 6= . 0 if k j is a basis of V0. Thus ϕ1, . . . , ϕn is the dual basis of the n standard basis e1,..., en of F . Example: Define D: P(R) → P(R) by Dp = p0. Suppose ϕ is the linear functional on P(R) defined by ϕ(p) = p(3). Then D0(ϕ) is the linear functional on P(R) given by D0(ϕ)
(p) = (ϕ ◦ D)(p) = ϕ(Dp) = ϕ(p0) = p0(3). Thus D0(ϕ) is the linear functional on P(R) that takes p to p0(3). R 1 Suppose ϕ is the linear functional on P(R) defined by ϕ(p) = 0 p. Then D0(ϕ) is the linear functional on P(R) given by Z 1 D0(ϕ)
(p) = (ϕ ◦ D)(p) = ϕ(Dp) = ϕ(p0) = p0 = p(1) − p(0). 0 Thus D0(ϕ) is the linear functional on P(R) that takes p to p(1) − p(0).

The Dual Map

Definition: dual map, T0

If T ∈ L(V, W), then the dual map of T is the linear map T0 ∈ L(W0, V0) defined by T0(ϕ) = ϕ ◦ T for ϕ ∈ W0. Suppose ϕ is the linear functional on P(R) defined by ϕ(p) = p(3). Then D0(ϕ) is the linear functional on P(R) given by D0(ϕ)
(p) = (ϕ ◦ D)(p) = ϕ(Dp) = ϕ(p0) = p0(3). Thus D0(ϕ) is the linear functional on P(R) that takes p to p0(3). R 1 Suppose ϕ is the linear functional on P(R) defined by ϕ(p) = 0 p. Then D0(ϕ) is the linear functional on P(R) given by Z 1 D0(ϕ)
(p) = (ϕ ◦ D)(p) = ϕ(Dp) = ϕ(p0) = p0 = p(1) − p(0). 0 Thus D0(ϕ) is the linear functional on P(R) that takes p to p(1) − p(0).

The Dual Map

Definition: dual map, T0

If T ∈ L(V, W), then the dual map of T is the linear map T0 ∈ L(W0, V0) defined by T0(ϕ) = ϕ ◦ T for ϕ ∈ W0.

Example: Define D: P(R) → P(R) by Dp = p0. Then D0(ϕ) is the linear functional on P(R) given by D0(ϕ)
(p) = (ϕ ◦ D)(p) = ϕ(Dp) = ϕ(p0) = p0(3). Thus D0(ϕ) is the linear functional on P(R) that takes p to p0(3). R 1 Suppose ϕ is the linear functional on P(R) defined by ϕ(p) = 0 p. Then D0(ϕ) is the linear functional on P(R) given by Z 1 D0(ϕ)
(p) = (ϕ ◦ D)(p) = ϕ(Dp) = ϕ(p0) = p0 = p(1) − p(0). 0 Thus D0(ϕ) is the linear functional on P(R) that takes p to p(1) − p(0).

The Dual Map

Definition: dual map, T0

If T ∈ L(V, W), then the dual map of T is the linear map T0 ∈ L(W0, V0) defined by T0(ϕ) = ϕ ◦ T for ϕ ∈ W0.

Example: Define D: P(R) → P(R) by Dp = p0. Suppose ϕ is the linear functional on P(R) defined by ϕ(p) = p(3). Thus D0(ϕ) is the linear functional on P(R) that takes p to p0(3). R 1 Suppose ϕ is the linear functional on P(R) defined by ϕ(p) = 0 p. Then D0(ϕ) is the linear functional on P(R) given by Z 1 D0(ϕ)
(p) = (ϕ ◦ D)(p) = ϕ(Dp) = ϕ(p0) = p0 = p(1) − p(0). 0 Thus D0(ϕ) is the linear functional on P(R) that takes p to p(1) − p(0).

The Dual Map

Definition: dual map, T0

If T ∈ L(V, W), then the dual map of T is the linear map T0 ∈ L(W0, V0) defined by T0(ϕ) = ϕ ◦ T for ϕ ∈ W0.

Example: Define D: P(R) → P(R) by Dp = p0. Suppose ϕ is the linear functional on P(R) defined by ϕ(p) = p(3). Then D0(ϕ) is the linear functional on P(R) given by D0(ϕ)
(p) = (ϕ ◦ D)(p) = ϕ(Dp) = ϕ(p0) = p0(3). R 1 Suppose ϕ is the linear functional on P(R) defined by ϕ(p) = 0 p. Then D0(ϕ) is the linear functional on P(R) given by Z 1 D0(ϕ)
(p) = (ϕ ◦ D)(p) = ϕ(Dp) = ϕ(p0) = p0 = p(1) − p(0). 0 Thus D0(ϕ) is the linear functional on P(R) that takes p to p(1) − p(0).

The Dual Map

Definition: dual map, T0

If T ∈ L(V, W), then the dual map of T is the linear map T0 ∈ L(W0, V0) defined by T0(ϕ) = ϕ ◦ T for ϕ ∈ W0.

Example: Define D: P(R) → P(R) by Dp = p0. Suppose ϕ is the linear functional on P(R) defined by ϕ(p) = p(3). Then D0(ϕ) is the linear functional on P(R) given by D0(ϕ)
(p) = (ϕ ◦ D)(p) = ϕ(Dp) = ϕ(p0) = p0(3). Thus D0(ϕ) is the linear functional on P(R) that takes p to p0(3). Then D0(ϕ) is the linear functional on P(R) given by Z 1 D0(ϕ)
(p) = (ϕ ◦ D)(p) = ϕ(Dp) = ϕ(p0) = p0 = p(1) − p(0). 0 Thus D0(ϕ) is the linear functional on P(R) that takes p to p(1) − p(0).

The Dual Map

Definition: dual map, T0

If T ∈ L(V, W), then the dual map of T is the linear map T0 ∈ L(W0, V0) defined by T0(ϕ) = ϕ ◦ T for ϕ ∈ W0.

Example: Define D: P(R) → P(R) by Dp = p0. Suppose ϕ is the linear functional on P(R) defined by ϕ(p) = p(3). Then D0(ϕ) is the linear functional on P(R) given by D0(ϕ)
(p) = (ϕ ◦ D)(p) = ϕ(Dp) = ϕ(p0) = p0(3). Thus D0(ϕ) is the linear functional on P(R) that takes p to p0(3). R 1 Suppose ϕ is the linear functional on P(R) defined by ϕ(p) = 0 p. Thus D0(ϕ) is the linear functional on P(R) that takes p to p(1) − p(0).

The Dual Map

Definition: dual map, T0

If T ∈ L(V, W), then the dual map of T is the linear map T0 ∈ L(W0, V0) defined by T0(ϕ) = ϕ ◦ T for ϕ ∈ W0.

Example: Define D: P(R) → P(R) by Dp = p0. Suppose ϕ is the linear functional on P(R) defined by ϕ(p) = p(3). Then D0(ϕ) is the linear functional on P(R) given by D0(ϕ)
(p) = (ϕ ◦ D)(p) = ϕ(Dp) = ϕ(p0) = p0(3). Thus D0(ϕ) is the linear functional on P(R) that takes p to p0(3). R 1 Suppose ϕ is the linear functional on P(R) defined by ϕ(p) = 0 p. Then D0(ϕ) is the linear functional on P(R) given by Z 1 D0(ϕ)
(p) = (ϕ ◦ D)(p) = ϕ(Dp) = ϕ(p0) = p0 = p(1) − p(0). 0 The Dual Map

Definition: dual map, T0

If T ∈ L(V, W), then the dual map of T is the linear map T0 ∈ L(W0, V0) defined by T0(ϕ) = ϕ ◦ T for ϕ ∈ W0.

Example: Define D: P(R) → P(R) by Dp = p0. Suppose ϕ is the linear functional on P(R) defined by ϕ(p) = p(3). Then D0(ϕ) is the linear functional on P(R) given by D0(ϕ)
(p) = (ϕ ◦ D)(p) = ϕ(Dp) = ϕ(p0) = p0(3). Thus D0(ϕ) is the linear functional on P(R) that takes p to p0(3). R 1 Suppose ϕ is the linear functional on P(R) defined by ϕ(p) = 0 p. Then D0(ϕ) is the linear functional on P(R) given by Z 1 D0(ϕ)
(p) = (ϕ ◦ D)(p) = ϕ(Dp) = ϕ(p0) = p0 = p(1) − p(0). 0 Thus D0(ϕ) is the linear functional on P(R) that takes p to p(1) − p(0). (S + T)0 = S0 + T0 for all S, T ∈ L(V, W). (λT)0 = λT0 for all λ ∈ F and all T ∈ L(V, W). (ST)0 = T0S0 for all T ∈ L(U, V) and all S ∈ L(V, W).

Algebraic Properties of Dual Maps

Algebraic properties of dual maps (λT)0 = λT0 for all λ ∈ F and all T ∈ L(V, W). (ST)0 = T0S0 for all T ∈ L(U, V) and all S ∈ L(V, W).

Algebraic Properties of Dual Maps

Algebraic properties of dual maps

(S + T)0 = S0 + T0 for all S, T ∈ L(V, W). (ST)0 = T0S0 for all T ∈ L(U, V) and all S ∈ L(V, W).

Algebraic Properties of Dual Maps

Algebraic properties of dual maps

(S + T)0 = S0 + T0 for all S, T ∈ L(V, W). (λT)0 = λT0 for all λ ∈ F and all T ∈ L(V, W). Algebraic Properties of Dual Maps

Algebraic properties of dual maps

(S + T)0 = S0 + T0 for all S, T ∈ L(V, W). (λT)0 = λT0 for all λ ∈ F and all T ∈ L(V, W). (ST)0 = T0S0 for all T ∈ L(U, V) and all S ∈ L(V, W). Linear Algebra Done Right, by Sheldon Axler