VECTOR SPACES ELECTRONIC VERSION OF LECTURE Dr Lê Xuân Đại HoChiMinh City University of Technology Faculty of Applied Science, Department of Applied Mathematics Email: ytkadai@hcmut.edu.vn HCMC — 2018 Dr Lê Xuân Đại (HCMUT-OISP) VECTOR SPACES HCMC — 2018 / 49 OUTLINE VECTOR SPACE AXIOMS LINEAR INDEPENDENCE AND DEPENDENCE SPANNING SET AND BASIS COORDINATES RELATIVE TO A BASIS Dr Lê Xuân Đại (HCMUT-OISP) VECTOR SPACES HCMC — 2018 / 49 Vector Space Axioms The set of real numbers + : R×R → R (x, y) → x + y •:R→R (λ, x) → λ.x Real Vector Spaces Polynomials of Degree n + : P n (x) × P n (x) → P n (x) (p(x), q(x)) → p(x) + q(x) • : R × P n (x) → P n (x) (λ, p(x)) → λ.p(x) The set of complex numbers + : C×C → C (x, y) → x + y VECTOR SPACES •:C→C (λ, x) → λ.x Dr Lê Xuân Đại (HCMUT-OISP) VECTOR SPACES HCMC — 2018 / 49 Vector Space Axioms Real Vector Spaces Vectors in 2-Space + : R2 × R2 → R2 → − → − → − → − (x, y )→ x + y −−→ −−→ −−→ −−→ (OM, ON) OM + ON ã : R ì R2 R2 → − → − (λ, x ) → λ x −−→ −−→ (λ, OM) → λ.OM Vectors in 3-Space + : R3 × R3 → R3 → − → − → − → − (x, y )→ x + y −−→ −−→ −−→ −−→ (OM, ON) → OM + ON VECTOR SPACES ã : R ì R3 → R3 → − → − (λ, x ) → λ x −−→ −−→ (λ, OM) → λ.OM Dr Lê Xuân Đại (HCMUT-OISP) VECTOR SPACES HCMC — 2018 / 49 Vector Space Axioms Real Vector Spaces Let V = ∅ on which operations are defined: + : V ×V → V (x, y) −→ x + y ã : RìV V (, x) .x If the following axioms are satisfied by: ∀x, y, z ∈ V , ∀λ, µ ∈ R x+y = y+x (λ + µ)x = λx + µx x + (y + z) = (x + y) + z λ(x + y) = λx + λy ∃0 ∈ V : x + = x λ(µx) = (λ.µ)x ∃(−x) ∈ V : x + (−x) = 1.x = x then V is called real vector space Dr Lê Xuân Đại (HCMUT-OISP) VECTOR SPACES HCMC — 2018 / 49 Linear Independence and Dependence Linear combination of vectors DEFINITION 2.1 If w is a vector in a vector space V , then w is said to be a linear combination of the vectors v1 , v2 , , ∈ V , if w can be expressed in the form n w= λi vi = λ1 v1 + λ2 v2 + + λn , i=1 where λ1, λ2, , λn are scalars These scalars are called the coefficients of the linear combination Dr Lê Xuân Đại (HCMUT-OISP) VECTOR SPACES HCMC — 2018 / 49 Linear Independence and Dependence Linear combination of vectors SHOW THAT w IS A LINEAR COMBINATION OF v1 , v2 , , In order for w to be a linear combination of v1 , v2 , , , there must be scalars λ1 , λ2 , , λn such that w = λ1v1 + λ2v2 + + λnvn If this system is consistent then w is a linear combination of v1, v2, , If this system is inconsistent then w is NOT a linear combination of v1, v2, , Dr Lê Xuân Đại (HCMUT-OISP) VECTOR SPACES HCMC — 2018 / 49 Linear Independence and Dependence Linear combination of vectors EXAMPLE 2.1 Show that w = (1, 4, −3) is a linear combination of v1 = (2, 1, 1), v2 = (−1, 1, −1), v3 = (1, 1, −2) In order for w to be a linear combination of v1 , v2 , v3 , there must be scalars λ1 , λ2 , λ3 such that λ1 v1 + λ2 v2 + λ3 v3 = w ⇔ (2λ1 , λ1 , λ1 ) + (−λ2 , λ2 , −λ2 ) + (λ3 , λ3 , −2λ3 ) = (1, 4, −3) Dr Lê Xuân Đại (HCMUT-OISP) VECTOR SPACES HCMC — 2018 / 49 Linear Independence and Dependence Linear combination of vectors    2λ1 − λ2 + λ3 = λ1 + λ2 + λ3 = ⇔   λ − λ − 2λ = −3   3      −1 λ1  λ1 =      ⇔  1   λ2  =   ⇔ λ2 =   λ =1 −1 −2 λ3 −3 Therefore, w = (1, 4, −3) is a linear combination of v1 = (2, 1, 1), v2 = (−1, 1, −1), v3 = (1, 1, −2) and w = v1 + 2v2 + v3 Dr Lê Xuân Đại (HCMUT-OISP) VECTOR SPACES HCMC — 2018 / 49 Linear Independence and Dependence Linear combination of vectors EXAMPLE 2.2 Determine whether w = (4, 3, 5) is a linear combination of v1 = (1, 2, 5), v2 = (1, 3, 7), v3 = (−2, 3, 4) or not? In order for w to be a linear combination of v1 , v2 , v3 , there must be scalars λ1 , λ2 , λ3 such that λ1 v1 + λ2 v2 + λ3 v3 = w ⇔ (λ1 , 2λ1 , 5λ1 ) + (λ2 , 3λ2 , 7λ2 ) + (−2λ3 , 3λ3 , 4λ3 ) = (4, 3, 5) Dr Lê Xuân Đại (HCMUT-OISP) VECTOR SPACES HCMC — 2018 10 / 49 ... C×C → C (x, y) → x + y VECTOR SPACES •:C→C (λ, x) → λ.x Dr Lê Xuân Đại (HCMUT-OISP) VECTOR SPACES HCMC — 2018 / 49 Vector Space Axioms Real Vector Spaces Vectors in 2 -Space + : R2 × R2 → R2 →... real vector space Dr Lê Xuân Đại (HCMUT-OISP) VECTOR SPACES HCMC — 2018 / 49 Linear Independence and Dependence Linear combination of vectors DEFINITION 2.1 If w is a vector in a vector space. .. (λ, x ) → λ x −−→ −−→ (λ, OM) → λ.OM Dr Lê Xuân Đại (HCMUT-OISP) VECTOR SPACES HCMC — 2018 / 49 Vector Space Axioms Real Vector Spaces Let V = ∅ on which operations are defined: + : V ×V → V (x,