The document I have just pasted it's a document or 747 kilobytes document, Study Guides, Projects, Research of Linear Algebra

Download The document I have just pasted it's a document or 747 kilobytes document and more Study Guides, Projects, Research Linear Algebra in PDF only on Docsity! LINEAR ALGEBRA I: TUTORIAL EXERCISE I (Vector Spaces) 1. Determine whether or not the given set is a vector space under the given operations. If it is not a vector space, list all properties that fail to hold. (a) Wo,s(IR), the set of all 2 by 3 matrices whose second column consists of zeros with usual matrix operations. (b) We(IR), the set of all 2 by 3 matrices the sum of whose entries is 1 with usual matrix operations. (c) Pa(IR), the set of all polynomials of degree less than or equal to 3 with usual polynomial operations. (a) The sct of all polynomials with positive coefficients. (c) The set 5 = {x,y} where x # y with addition defined by e+e aa, ety syryte ytye and scalar multiplication defined by ar=a ay =2 for all scalars a. 2. Determine whether or not the given set is a subspace of M2(IR), the vector space of all 2 by 2 matrices with real entries. (a) The set of all 2 by 2 matrices the sum of whose entries is zero. (b) the set of all 2by 2 matrices whose determinant is zero. b (c) uesss={(* : ance} el (a) The set $= 4 ( ° :a,beR b 2a+b 3. Determine whether or not cach of the following sets of vectors of IR? is linearly independent. (a) {(1,2,-1), (8,1, -2)} (b) {(1,3,2), (2,1,0), (0,5, 4)} (c) {(2,-1,1), (1,2,3), (0, 1,2)} (a) {(1,0,—1), (21,3), (1,0,0), (1,01)} be) CamScanner