SUBSPACES 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) SUBSPACES HCMC — 2018 / 41 OUTLINE SUBSPACES OPERATIONS WITH SUBSPACES Dr Lê Xuân Đại (HCMUT-OISP) SUBSPACES HCMC — 2018 / 41 Subspaces Definition SUBSPACES OF R2 : LINES THROUGH THE ORIGIN Dr Lê Xuân Đại (HCMUT-OISP) SUBSPACES HCMC — 2018 / 41 Subspaces Definition SUBSPACES OF R3 : LINES THROUGH THE ORIGIN Dr Lê Xuân Đại (HCMUT-OISP) SUBSPACES HCMC — 2018 / 41 Subspaces Definition SUBSPACES OF R3 : PLANES THROUGH THE ORIGIN Dr Lê Xuân Đại (HCMUT-OISP) SUBSPACES HCMC — 2018 / 41 Subspaces Definition A SUBSET OF R3 THAT IS NOT A SUBSPACE Dr Lê Xuân Đại (HCMUT-OISP) SUBSPACES HCMC — 2018 / 41 Subspaces Definition A SUBSET OF R3 THAT IS NOT A SUBSPACE Dr Lê Xuân Đại (HCMUT-OISP) SUBSPACES HCMC — 2018 / 41 Subspaces Definition DEFINITION 1.1 A subset W of a vector space V is called a subspace of V if and only if the following conditions are satisfied W =∅ ∀x, y ∈ W , x + y ∈ W ∀λ ∈ R, ∀x ∈ W , λx ∈ W Dr Lê Xuân Đại (HCMUT-OISP) SUBSPACES HCMC — 2018 / 41 Subspaces Definition EXAMPLE 1.1 W = R × {0} = {(x1 , x2 ) : x1 ∈ R, x2 = 0} is the subspace of R2 We have W ⊂ R2, (0, 0) ∈ W ⇒ W = ∅ For all x = (x1 , 0), y = (y1 , 0) ∈ W we have x + y = (x1 + y1 , 0) ∈ W , ∀λ ∈ R, λx = (λx1 , 0) ∈ W Therefore, W is the subspace of R2 Dr Lê Xuân Đại (HCMUT-OISP) SUBSPACES HCMC — 2018 / 41 Subspaces Definition EXAMPLE 1.2 W = {(x1 , x2 , x3 ) ∈ R3 : 2x1 − 2x2 + x3 = 0} is the subspace of R3 We have W ⊂ R3, (0, 0, 0) ∈ W ⇒ W = ∅ ∀x = (x1 , x2 , x3 ), y = (y1 , y2 , y3 ) ∈ W ⇒ 2x1 − 2x2 + x3 = 0, 2y1 − 2y2 + y3 = Therefore, x + y = (x1 + y1, x2 + y2, x3 + y3), and 2(x1 + y1 ) − 2(x2 + y2 ) + (x3 + y3 ) = (2x1 − 2x2 + x3 ) + (2y1 − 2y2 + y3 ) = ⇒ x + y ∈ W Dr Lê Xuân Đại (HCMUT-OISP) SUBSPACES HCMC — 2018 10 / 41 Operations with Subspaces A basis of the intersection and the sum of subspaces FORMULA CONNECTING THE DIMENSIONS OF THE SUM AND THE INTERSECTION OF SUBSPACES THEOREM 2.3 Let U and W be subspaces of a finitely generated vector space V Then dim(U + W ) = dim(U) + dim(W ) − dim(U ∩ W ) Dr Lê Xuân Đại (HCMUT-OISP) SUBSPACES HCMC — 2018 27 / 41 Operations with Subspaces A basis of the intersection and the sum of subspaces EXAMPLE 2.1 In R3 the following vectors u1 = (1, 2, 1), u2 = (3, 6, 5), u3 = (4, 8, 6), u4 = (8, 16, 12) and w1 = (1, 3, 3), w2 = (2, 5, 5), w3 = (3, 8, 8), w4 = (6, 16, 16) are given Let U = span u1 , u2 , u3 , u4 , W = span w1 , w2 , w3 , w4 Find a basis and dimension of the sum U + W and the intersection U ∩ W Dr Lê Xuân Đại (HCMUT-OISP) SUBSPACES HCMC — 2018 28 / 41 Operations with Subspaces A basis of the intersection and the sum of subspaces The basis and dimension of U  3   4 8 16 12    0   →  0 0 0      Therefore, dim(U) = and the basis of U is (1, 2, 1), (3, 6, 5) Dr Lê Xuân Đại (HCMUT-OISP) SUBSPACES HCMC — 2018 29 / 41 Operations with Subspaces A basis of the intersection and the sum of subspaces The basis and dimension of W  2   3 16 16     −1   →  0 0 −1 0      Therefore, dim(W ) = and the basis of W is (1, 3, 3), (2, 5, 5) Dr Lê Xuân Đại (HCMUT-OISP) SUBSPACES HCMC — 2018 30 / 41 Operations with Subspaces A basis of the intersection and the sum of subspaces THE BASIS AND DIMENSION OF THE SUM U + W The sum U + W is spanned by the vectors (1, 2, 1), (3, 6, 5), (1, 3, 3), (2, 5, 5)    1 3 5 0     → 1 3 0 2 5 0      Therefore, dim(U + W ) = and the basis of U + W is (1, 2, 1), (3, 6, 5), (1, 3, 3) Dr Lê Xuân Đại (HCMUT-OISP) SUBSPACES HCMC — 2018 31 / 41 Operations with Subspaces A basis of the intersection and the sum of subspaces THE BASIS AND DIMENSION OF U ∩ W x = α1 (1, 2, 1) + α2 (3, 6, 5) x = α3 (1, 3, 3) + α4 (2, 5, 5) x ∈ U ∩W ⇔ ⇒ α1 (1, 2, 1) + α2 (3, 6, 5) = = α3 (1, 3, 3) + α4 (2, 5, 5) ⇔ α3 = −α4 = −2α2 = 2α1 ⇒ x = α1 (1, 2, 1) + α2 (3, 6, 5) = α2 (2, 4, 4) Therefore, dim(U ∩ W ) = and the basis of U ∩ W is (2, 4, 4) Dr Lê Xuân Đại (HCMUT-OISP) SUBSPACES HCMC — 2018 32 / 41 Operations with Subspaces A basis of the intersection and the sum of subspaces EXAMPLE 2.2 For R3 the subspaces U = (x1 , x2 , x3 ) x1 + x2 − 2x3 = x1 − x2 − 2x3 = and W = (x1 , x2 , x3 ) : x1 = x2 are given Find a basis and dimension of U + W and U ∩ W Dr Lê Xuân Đại (HCMUT-OISP) SUBSPACES HCMC — 2018 33 / 41 Operations with Subspaces A basis of the intersection and the sum of subspaces The basis and dimension of U 1 −2 1 −2 → −1 −2 −2 Therefore, dim(U) = and the basis of U is (2, 0, 1) The basis and dimension of W −1 Therefore, dim(W ) = and the basis of W is (1, 1, 0), (0, 0, 1) Dr Lê Xuân Đại (HCMUT-OISP) SUBSPACES HCMC — 2018 34 / 41 Operations with Subspaces A basis of the intersection and the sum of subspaces THE BASIS AND DIMENSION OF THE SUM U + W The sum U + W is spanned by the vectors (2, 0, 1), (1, 1, 0), (0, 0, 1)     1      1  →  −2  0 0 Therefore, dim(U + W ) = and the basis of U + W is (2, 0, 1), (1, 1, 0), (0, 0, 1) Dr Lê Xuân Đại (HCMUT-OISP) SUBSPACES HCMC — 2018 35 / 41 Operations with Subspaces A basis of the intersection and the sum of subspaces THE BASIS AND DIMENSION OF THE INTERSECTION U ∩W    x1 + x2 − 2x3 = x ∈ U ∩ W ⇔ x1 − x2 − 2x3 =   x =x ⇔ x1 = x2 = x3 = Therefore, dim(U ∩ W ) = and there is no basis of U ∩ W Dr Lê Xuân Đại (HCMUT-OISP) SUBSPACES HCMC — 2018 36 / 41 Operations with Subspaces A basis of the intersection and the sum of subspaces EXAMPLE 2.3 For R3, the subspaces U = (x1 , x2 , x3 ) ∈ R3 : x1 + x2 + x3 = , and W = span (1; 0; 1), (2; 3; 1) are given Find a basis and dimension of U ∩ W and U + W Dr Lê Xuân Đại (HCMUT-OISP) SUBSPACES HCMC — 2018 37 / 41 Operations with Subspaces A basis of the intersection and the sum of subspaces The basis and dimension of U x1 + x2 + x3 = ⇒ x1 = −x2 − x3 Therefore, dim(U) = and the basis of U is (−1, 1, 0), (−1, 0, 1) The basis and dimension of W 1 1 → 3 −1 Therefore, dim(W ) = and the basis of W is (1, 0, 1), (2, 3, 1) Dr Lê Xuân Đại (HCMUT-OISP) SUBSPACES HCMC — 2018 38 / 41 Operations with Subspaces A basis of the intersection and the sum of subspaces THE BASIS AND DIMENSION OF THE SUM U + W The sum U + W is spanned by the vectors (−1, 1, 0), (−1, 0, 1), (1, 0, 1), (2, 3, 1)    −1 −1  −1   −1     →  1  0 0      Therefore, dim(U + W ) = and the basis of U + W is (−1, 1, 0), (−1, 0, 1), (1, 0, 1) Dr Lê Xuân Đại (HCMUT-OISP) SUBSPACES HCMC — 2018 39 / 41 Operations with Subspaces A basis of the intersection and the sum of subspaces THE BASIS AND DIMENSION OF U ∩ W ∀x ∈ U ∩ W ⇐⇒ x ∈ U ∧ x ∈ W x ∈ W ⇐⇒ x = α(1, 0, 1) + β(2, 3, 1) ⇐⇒ x = (α + 2β, 3β, α + β) x ∈ U ⇐⇒ x satisfies the condition of U : α + 2β + 3β + α + β = ⇐⇒ α = −3β x = (α + 2β, 3β, α + β) = = (−β, 3β, −2β) = β(−1, 3, −2) Therefore, (−1, 3, −2) is the basis of U ∩ W and dim(U ∩ W ) = Dr Lê Xuân Đại (HCMUT-OISP) SUBSPACES HCMC — 2018 40 / 41 Operations with Subspaces A basis of the intersection and the sum of subspaces THANK YOU FOR YOUR ATTENTION Dr Lê Xuân Đại (HCMUT-OISP) SUBSPACES HCMC — 2018 41 / 41 ...OUTLINE SUBSPACES OPERATIONS WITH SUBSPACES Dr Lê Xuân Đại (HCMUT-OISP) SUBSPACES HCMC — 2018 / 41 Subspaces Definition SUBSPACES OF R2 : LINES THROUGH THE ORIGIN Dr Lê Xuân Đại (HCMUT-OISP) SUBSPACES. .. SUBSPACES HCMC — 2018 / 41 Subspaces Definition SUBSPACES OF R3 : LINES THROUGH THE ORIGIN Dr Lê Xuân Đại (HCMUT-OISP) SUBSPACES HCMC — 2018 / 41 Subspaces Definition SUBSPACES OF R3 : PLANES THROUGH... Lê Xuân Đại (HCMUT-OISP) SUBSPACES HCMC — 2018 / 41 Subspaces Definition A SUBSET OF R3 THAT IS NOT A SUBSPACE Dr Lê Xuân Đại (HCMUT-OISP) SUBSPACES HCMC — 2018 / 41 Subspaces Definition A SUBSET