that are continuous on [-1,1 28. The set of all functions such that f(0) 1 Subse
ID: 3115669 • Letter: T
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that are continuous on [-1,1 28. The set of all functions such that f(0) 1 Subsets That Are Not Subspaces In Exercises 7-20, Determining Subspaces In Exercises 29-36, W is not a subspace of the vector space. Verify this by determine whether the subset of M,. is a subspace of giving a specific example that violates the test for a M with the standard operations of matrix addition vector subspace (Theorem 4.5) 7, W is the set of all vectors in R3 whose third component 29. The set of all n × n upper triangular matrices and scalar multiplication. 30. The set of all n × n diagonal matrices 8. W is the set of all vectors in R2 whose second The set of all n × n matrices with integer entries The set of all n × n matrices A that commute with a given matrix B; that is, AB -BA component is 1. 3 9. W is the set of all vectors in R2 whose components are ational numbers. is the set of all vectors in R2 whose components are integers. 33. The set of all n × n singular matrices 34. The set of all n × n invertible matrices 11. Wis the set of all nonnegative functions in C-0, 0).35. The set of all n x n matrices whose entries add up to zero 36, The set of all n × n matrices whose trace is nonzero 12. W is the set of all linear functions ax + b, a 13. W is the set of all vectors in R3 whose components are 14 W is the set of all vectors in R3 whose components are 15. W is the set of all matrices in M3,3 of the form 0, in C(-co, oo). nonnegative. Pythagorean triples. Determining Subspaces In Exercises 37-42, determine whether the set W is a subspace of R3 with the standard operations. Justify your answer 37. W [Cx,x2, 0): x, and x2 are real numbers) 38. W = {(x1, x2, 4): x1 and x2 are real numbers} 39. W-((a, b, a + 2b): a and b are real numbers) 40. W = {(s, s-1,1): s and t are real numbers} 41. W = { (xi, X2,X1X2): Xi and x2 are real numbers) 42. W = {(x1, 1/x1, X31 and x3 are real numbers, xi c 0 d 0)Explanation / Answer
10. We have W = { (x,y)T: x,y R and x,y are integers}. Now, if be a non-integer scalar, then (x,y)T= (x,y)T may not belong to W. For example, 1/3(2,5)T = (2/3,5/3)T W. Hence W is not closed under scalar multiplication, and therefore, W is not a subspace of R2.
14. Here, W = {(a,b,c)T: a,b,c R and c2 = a2+b2}. Now, if X = (3,4,5)T and Y = (5,12,13)T, then X,Y W. Further, X+Y = (3,4,5)T +(5,12,13)T = (8,16,18)T W as 182 82+162. Hence W is not closed under vector addition, and therefore, W is not a subspace of R3.
32. Here, W = { A: A Mnxn and AB = BA}. Let A and C be 2 arbitrary elements of W and let be an arbitrary scalar. Then AB = BA and CB = BC. Further, (A+C)B = AB +CB = BA+BC = B(A+C) which shows that A+C W so that W is closed under vector addition. Also (A)B = (AB) = (BA)= B(A). Hence A W so that W is closed under scalar multiplication. Further, the nxn zero matrix 0 commutes with all nxn matrices so that 0 W. Hence W is a vector space, and therefore, W is a subspace of Mnxn.
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