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Introduction to Linear Algebra (Classic Version) (5th Edition) (Pearson Modern Classics for Advanced Mathematics Series)
- In Exercises 24-45, use Theorem 6.2 to determine whether W is a subspace of V. V=Mnn,WAinMnn:detA=1arrow_forwardAn mΓn matrix A is called upper triangular if all entries lying below the diagonal entries are zero, that is, if Aij= 0 whenever i > j. Prove that the upper triangular matrices form a subspace of MmΓ n(F ).arrow_forwardLet P be the projection matrix corresponding to a subspace S of Rm. Show that P2 = Parrow_forward
- 4. Consider the following subspaces of P.H = Span{1 + t, 1 β t3} and G = Span{1 + t + t2, t β t3, 1 + t + t3}Find dim H, dim G and dim H β© G.arrow_forwardDetermine whether the following are subspaces of C[β1, 1]: The set of odd functions in C[β1, 1]arrow_forwardLet P2 denote the vector space of polynomials of degree up to 2. Which of the following subsets of P2 are subspaces of P2?arrow_forward
- In linear algebra, a subspace of R^n can have a dimension less than n.Β (True or False)arrow_forwardIn which of the following examples is the set U not a subspace of the space V?arrow_forwardFind the companion matrix of p(x) = x2 -7x + 12 and then find the characteristic polynomial of C( p).arrow_forward
- can a subspace of R^n have a dimension less than n.arrow_forwardUse Theorem 4.2.1 to determine which of the following are subspaces of Mnn. (a) The set of all diagonal n Γ n matrices. (b) The set of all n Γ n matrices A such that det(A) = 0. (c) ThesetofallnΓnmatricesAsuchthattr(A)=0. (d) The set of all symmetric n Γ n matrices. (e) ThesetofallnΓnmatricesAsuchthatAT =βA. (f) ThesetofallnΓnmatricesAforwhichAx = 0hasonly the trivial solution. (g) ThesetofallnΓnmatricesAsuchthatAB=BAfor some fixed n Γ n matrix B.arrow_forwardFind the companion matrix of p(x) = x3 + 3x2 -4x + 12 and then find the characteristic polynomial of C(p)arrow_forward
- Linear Algebra: A Modern IntroductionAlgebraISBN:9781285463247Author:David PoolePublisher:Cengage Learning