Dr. Abdulla Eid. Section 4.2 Subspaces. Dr. Abdulla Eid. MATHS 211: Linear Algebra. College of Science
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1 Section 4.2 Subspaces College of Science MATHS 211: Linear Algebra (University of Bahrain) Subspaces 1 / 42
2 Goal: 1 Define subspaces. 2 Subspace test. 3 Linear Combination of elements. 4 Subspace generated by elements (Span). (University of Bahrain) Subspaces 2 / 42
3 Subspace Definition 1 Let V be a vector space. A subset W of V is called a subspace of V if W is itself a vector space under the same operations of V. (University of Bahrain) Subspaces 3 / 42
4 Subspace Test Theorem 2 If W is a subset of V such that 1 0 W. 2 For all u, v W, we have u + v W. (Closed under addition Axiom (1) ) 3 For all u W, k R, we have ku W. (Closed under scalar multiplication Axiom (6) ) Then W is a subspace of V. In short, we need to check that the zero is in W and W is closed under + and (University of Bahrain) Subspaces 4 / 42
5 Zero Subspace Example 3 Let V be any vector space. Let W = {0}. Then W is a subspace of V. We call W the zero subspace of V. (University of Bahrain) Subspaces 5 / 42
6 Lines through the origin Example 4 Let m be a fixed real number. Consider the subset of V = R 2 {( ) } x W := x R mx Then W is a subspace of R 2 (University of Bahrain) Subspaces 6 / 42
7 Example 5 Determine whether the following is a subspace of R 3 or not. W := a b a, b, c R, c = a b c (University of Bahrain) Subspaces 7 / 42
8 Example 6 Determine whether the following is a subspace of R 3 or not. W := a 1 a R, 0 (University of Bahrain) Subspaces 8 / 42
9 Example 7 Determine whether the following is a subspace of Mat(n, n, R) or not. W := { } A Mat(n, n, R) A T = A (University of Bahrain) Subspaces 9 / 42
10 Example 8 Determine whether the following is a subspace of Mat(n, n, R) or not. W := {A Mat(n, n, R) tr(a) = 0} (University of Bahrain) Subspaces 10 / 42
11 Example 9 Determine whether the following is a subspace of P 3 or not. W := { a 0 + a 1 X + a 2 X 2 + a 3 X 3 a 1 = a 2 } (University of Bahrain) Subspaces 11 / 42
12 Example 10 Determine whether the following is a subspace of Maps(R, R) or not. C (, ) := {f f is continuous } C 1 (, ) := {f f is differentiable } C 2 (, ) := {f f is twice differentiable } C (, ) := {f f is infinitely many differentiable } Smooth functions Give an example of a function in C (, )? (University of Bahrain) Subspaces 12 / 42
13 Intersection of subspaces Theorem 11 Let W 1, W 2 be two subspaces of a vector space V. Then, the intersection of W 1 and W 2 is also a subspace of V. Theorem 12 Let W 1, W 2,..., W n be two subspaces of a vector space V. Then, the intersection of W 1, W 2,..., W n is also a subspace of V. (University of Bahrain) Subspaces 13 / 42
14 Union of subspaces Example 13 Let W 1, W 2 be two subspaces of a vector space R 2 that are given by W 1 = {( ) } x x R, W 0 1 = Verify that W 1, W 2 are subspaces but W 1 W 2 is not. {( ) } 0 y R y (University of Bahrain) Subspaces 14 / 42
15 Do HOMEWORK 1 (University of Bahrain) Subspaces 15 / 42
16 Linear Combination Definition 14 If w is a vector in a vector space V, then w is said to be a linear combination of the vectors v 1,..., v n if w can be expressed in the form w = k 1 v 1 + k 2 v k n v n, where k 1, k 2,..., k n R which are called the coefficients of the linear combination. (University of Bahrain) Subspaces 16 / 42
17 Example 15 Express the following as linear combination of u = (2, 1, 4), v = (1, 1, 3), and w = (3, 2, 5). 1 (6, 11, 6) 2 (7, 8, 9) (University of Bahrain) Subspaces 17 / 42
18 Solution (University of Bahrain) Subspaces 18 / 42
19 Example 16 Let u = (1, 3, 2), v = (1, 0, 4). Determine whether the following is a linear combination of u and v. 1 (0, 3, 6) 2 (1, 6, 16) (University of Bahrain) Subspaces 19 / 42
20 Example 17 Express the following as linear combination of A = ( ) ( ) B =, and C = ( ) ( ) ( ) 3 2, 0 1 (University of Bahrain) Subspaces 20 / 42
21 Solution (University of Bahrain) Subspaces 21 / 42
22 Example 18 Express the following as linear combination of P 1 = 2 + X + 4x 2, P 2 = 1 X + 3x 2, and P 3 = 3 + 2X + 5X X + 6X 2 (University of Bahrain) Subspaces 22 / 42
23 Solution (University of Bahrain) Subspaces 23 / 42
24 Subset generated by elements is a subspace Theorem 19 If S = {v 1, v 2,..., v n } is a nonempty set of vectors in a vector space V. Then, 1 The set W of all possible linear combinations of vectors in S is a subspace, i.e., W = {k 1 v 1 + k 2 v k n v n k 1, k 2,..., k n R} 2 The set W is the smallest subspace of V that contain all of the vectors in S in the sense of containment relationship. We denote the set W above by W = span{v 1,..., v n } or W = span(s) or W = v 1,..., v n (University of Bahrain) Subspaces 24 / 42
25 Proof (University of Bahrain) Subspaces 25 / 42
26 Recall: The equation Ax = b Theorem 20 The following are equivalent: 1 A is invertible. 2 det(a) = 0. 3 The reduced row echelon form is I n. 4 Ax = b is consistent for every n 1 matrix b. 5 Ax = b has a unique solution for every n 1 matrix b. (University of Bahrain) Subspaces 26 / 42
27 Spanning set of the whole space Example 21 Determine whether v 1 = (2, 1, 2), v 2 = (4, 1, 3), and v 3 = (2, 2, 1) span R 3. (University of Bahrain) Subspaces 27 / 42
28 Solution (University of Bahrain) Subspaces 28 / 42
29 Spanning set of the whole space Example 22 Determine whether P 1 = 1 + X + X 2, P 2 = 3 + X, P 3 = 5 X + 4X 2, and P 4 = 2 2X + 2X 2 span P 2. (University of Bahrain) Subspaces 29 / 42
30 Solution (University of Bahrain) Subspaces 30 / 42
31 Standard spanning sets Note: The standard spanning set for R 2 is e 1, e 2, where e 1 = (1, 0) and e 2 = (0, 1) The standard spanning set for R 3 is e 1, e 2, e 3, where e 1 = (1, 0, 0), e 2 = (0, 1, 0) and e 3 = (0, 0, 1) The standard spanning set for R n is e 1, e 2, e 3,..., e n, where e 1 = (1, 0,..., 0), e 2 = (0, 1,..., 0), e 3 = (0, 0, 1, 0,..., 0) and e n = (0, 0,..., 1) The standard spanning set for P 2 is 1, X, X 2. The standard spanning set for P n is 1, X, X 2,..., X n. The standard spanning set for Mat(2, 2, R) is ( ) ( ) ( ) ( ) ,,,, (University of Bahrain) Subspaces 31 / 42
32 Spanning set of the subspace Example 23 Find a spanning set for the following subspace Let m be a fixed real number. Consider the subset of V = R 2 {( ) } x W := x R mx (University of Bahrain) Subspaces 32 / 42
33 Spanning set of the subspace Example 24 Find a spanning set for the following subspace W := a b a, b, c R, c = a b c (University of Bahrain) Subspaces 33 / 42
34 Spanning set of the subspace Example 25 Find a spanning set for the following subspace W := { } A Mat(n, n, R) A T = A (University of Bahrain) Subspaces 34 / 42
35 Spanning set of the subspace Example 26 Find a spanning set for the following subspace W := { a 0 + a 1 X + a 2 X 2 + a 3 X 3 } a 1 = a 2 (University of Bahrain) Subspaces 35 / 42
36 Null Space Theorem 27 Let A Mat(m, n, R) be a m n matrix. The subset W of R n defined by is a subspace of R n. W = {x Ax = 0} It is called the null space of A, denoted by Nul(A) and it is consisting of the solutions to the equation Ax = 0. (University of Bahrain) Subspaces 36 / 42
37 Example 28 1 Determine whether the w = 3 is in the null space of A = (University of Bahrain) Subspaces 37 / 42
38 Spanning set of the subspace Example 29 Find a spanning set for the null space of A = (University of Bahrain) Subspaces 38 / 42
39 Spanning set of the subspace Example 30 Find a spanning set for the null space of A = (University of Bahrain) Subspaces 39 / 42
40 Do HOMEWORK 1 (University of Bahrain) Subspaces 40 / 42
41 Equality of spanning sets Theorem 31 Let S = {v 1, v 2,..., v n } and S = {w 1, w 2,..., w n } be two sets of vectors. Then, span{v 1, v 2,..., v n } = span{w 1, w 2,..., w n } each v i is a linear combination of w i and each w i is a linear combination of the v i (University of Bahrain) Subspaces 41 / 42
42 Example 32 Show that span 1 0 0, = span 4 3 0, (University of Bahrain) Subspaces 42 / 42
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