MAT 240MAT 编程辅导、R程序设计写作

” MAT 240MAT 编程辅导、R程序设计写作MAT 240 MAT 240 MAT 240 – A SSIGNMENT SSIGNMENT SSIGNMENT # 4 D UE MARCH 21InstructionsPlease submit solutions to the questions listed below on Crowdmark. If you did not receive an emailfrom CM with this assignment, contact J. Thind. Your assignment is due before 11.59pm on Sunday,March 21.Full justification is required for all questions. Dont forget – proofs have words, not just strings ofcalculations/algebraic manipulations.Unless otherwise indicated, F denotes a field and V denotes a vector space over F.Assigned Questions(1) Let V = Sym2(F) (and assume that char(F) 6= 2).(a) Find Q – the change of coordinate matrix from to coordinates.(b) Find Q1- the change of coordinate matrix from to coordinates.(c) Find [T] and [T].(2) Let A Mnn(F) be a matrix with columns c1, . . . , cn.(a) Prove that the columns of A form a basis for Fnif and only if A is invertible.(b) Deduce that for every invertible matrix Q there exist bases , of Fnso that Q is thechange of coordinate matrix from to coordinates.(c) (T/F:) Suppose that V is any n-dimensional vector space and T : V V an isomorphism.If Q is invertible, then there exist bases , of V so that [T] = Q.(d) (T/F:) Suppose that V is any n-dimensional vector space and T : V V an isomorphism.If Q is invertible, then there exists a basis of V so that [T] = Q.Find the R – the RREF of A, and express R = QA for some invertiblematrix Q.(Hint: Keep track of your row Operations, and recall that every invertible matrix is a productof elementary matrices.)MAT 240 MAT 240 MAT 240 – A SSIGNMENT SSIGNMENT SSIGNMENT # 4 D UE MARCH 21(4) Prove that for every n n matrix A there exists an invertible matrix Q so that QA is uppertriangular.(Hint: Use induction on n and Row-operations/elementary matrices.)(5) Define an equivalence relation on Mmn(F) by A B if there exists invertible matricesQ Mmm(F), P Mnn(F) so that A = QBP.(a) Prove that this is an equivalence relation.(b) Express the number of distinct equivalence classes in Mmn(F) as a function of m and n.(And, as usual, prove your claim.)(6) (a) Let T : V W, S : W X Be isomorphisms. Prove that S T is an isomorphism andfind a formula for (S T)1.(b) Prove that the product of two invertible matrices of the same size is invertible using (a).(c) Give a second proof of the fact that the product of two invertible matrices of the samesize is invertible, using elementary matrices.(d) Give a third proof of the fact that the product of two invertible matrices of the same sizeis invertible, using rank.(7) Find an isomorphism T : Sk3(R) W, where W = {(x, y, z, w) R4| x + y + z + w = 0}.(As always, justify your claim!)(8) Let T : R2 R2 be a linear transformation So that T(1, 1) = (2, 2) and T(1, 0) = (1, 0). Let = {(1, 1),(1, 0)} and be the standard basis.(a) Find [T].(b) Find Q – the change of coordinate matrix from to coordinates.(c) Find Q1 using row operations on (Q|I2).(d) Find [T] and deduce an explicit formula for T(x, y).(e) Which coordinates are best suited for studying this transformation – or ? Why?(9) Determine if the statements below are true or false. If true, give a proof. If false, explain why,and/or provide a counterexample.(a) There exists a m n matrix A So that the system Ax = b has no solutions for all b Fm.(b) Suppose that A Mnn(F). If Ax = b has solutions for all b Fn, then A is invertible.(c) If A Mmn(F) has rank n 1, then Ax = b has no solutions for some b Fm.请加QQ：99515681 或邮箱：99515681@qq.com WX：codehelp

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