Let L Be An N N Elementary Unit Lower Triangular Matrix That Is L Has 1 S On The
/in Uncategorized /by developerLet L be an n × n elementary unit lower triangular matrix, that is, L has 1’s on the maindiagonal, and zeros in all other positions except column k. The nonzero entries in column kappear in positions k to n (and the (k, k) entry is, as already mentioned, equal to 1). Assumethe following equation holds,P1*P2 · · · Pq*L = M*P1*P2 · · · Pq,where each Pi, i = 1, . . . q is a permutation matrix that encodes an exchange between rows liand mi, such that k + 1 <= li <= n and k + 1 <= mi <= n, and M is some other n × n matrix.Show that if the above equation holds, then M must also be an elementary lower triangularmatrix all of whose nonzero entries are either 1’s on the diagonal or lie in positions k to n ofcolumn k. Furthermore, show that the entries of column k of M must be precisely the entriesof column k of L after the q swaps given by Pq, . . . , P1 are applied to this column. In otherwords, show thatM(:, k) = P1 · · · PqL(:, k).Hint: Let Q = P1 *P2 · · · Pq. Write Q in block-matrix form, with two diagonal blocks of size k×kand (n − k) × (n − k) respectively and corresponding off-diagonal block. The first diagonalblock is known in closed form. Why? The two off-diagonal blocks are also known in closedform. Why? Use block matrix-multiplication to separately evaluate columns 1 : k and alsocolumns k + 1 : n of the left-hand side Q*L and the right-hand side M*Q and determine theform of M.
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Let P 1 P 400 Be A Sample Computed By P X Y 2 Z 3 W 4 Where X 1 X 2 X 400 Y 1 Y
/in Uncategorized /by developerP = X + Y^2 + Z^3 + W^4, where
X_1, X_2, .., X_400Y_1, Y_2, .., Y_400Z_1, Z_2, .., Z_400W_1, W_2, .., W_400
are samples from Uniform(0,1) distribution. Find 90% confidence interval for the variance of P. Note: Use at least fifty thousand simulated samples to generate sampling distr. for the variance of P.
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Let L Be A Set In Three Dimensional Space And Let Lil I Be Equal To The Volume
/in Uncategorized /by developerPlease help me solve those questions, the questions are shown in the attachments (6, 7, 8)
- Attachment 1
- Attachment 2
5. LET L be a set in three – dimensional space and let LIL’I be equal to the Volume of’ `. if I has finite*volume . otherwise , let !It’s be undefined . Find Fizi .7 . For every one – dimensional set !’ for which the integral exists . Citi = To fluid , where flaj -Gril – Il, O _ _ _ I, zero elsewhere ; otherwise , let fic"; he undefined . Find Dic").16 ) [ 2 = 1 1 / 2) .8. Suppose the experiment is to choose a real number at random in the interval 10 . 1 1 . For any subinterval` , ` ) CID, I), it seems reasonable to assign the probablybability ?" [`, “] = 6 – a; Le, the probability ofselectin the point from the subinterval is directly proportional to the length of the subinterval . If thisis the case . choose an appropriate sequence of subintervals to shoe that Fall = " For all at 10. 1] .Hint : let It be a decreasing sequence of events , as in Question ? , then lim ,_ ta Pie! ! =Pllimn_ too Caj = Pill _; Cal
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Let P4 Be The Set Of All Polynomials With The Degree At Most 4 I P4 Contains All
/in Uncategorized /by developerLet P4 be the set of all polynomials with the degree at most 4, i.e., P4 contains all polynomials of the formp(x) = a4x4 + a3x3 + a2x2 + a1x + a0.- Show that P4 is a vector space.
Let P4 be the set of all polynomials with the degree at most 4, i.e., P4 contains all polynomials of the formp(x) = a4x4 + a3x3 + a2x2 + a1x + a0. ANSWER : , where
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Let L A And M Hx I Be Two Linear Mapping S A Prove That Rangel L O M Range The
/in Uncategorized /by developerCan anyone help me with this question please? The question 3 a) in assignment 6 is in the picture below.
Thank you so much!
- Attachment 1
- Attachment 2
3. Let L . A" ~ {‘ and M. HX _ I’ be two linear mapping’s( a ) Prove that Rangel L o M { Range [ ] . Then , conclude that Rank [ [ O AI < Rank [ [ ) (using Question Bla) ofAssignment 6 ).( b ) Prove that Mulling = NullIL – MI) . Conclude that Nullitying = Nullity [ [ O MJ (using Question 3 ( a) ofAssignment 6 ) . Then , using the Rank – Nullity theorem , conclude that Rank ! [ . MY = Ranking .( C ) Conclude that Rank [ [ O MY = min / Rank [ [ ) , Rank[ My] .(d ) Conclude that for an m* n matrix A and an n* * matrix B , we always have** COLLAB ) = COLLAY ;* Null( B ) = NullL ABY ; and* Rank ! ABY = min ( Rank[ A] , Rank [ BY] .Give an example for matrices A and B such that Rank [ AB ) = min ( Rank [ A ] , Rank [ BY] , and an example*for matrices C and D such that Rank ! (D ) < min ( Rank![ ] , Rank [ D ]1 .
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Let P2 Denote The Vector Space Of All Polynomials With Real Coefcients And Of De
/in Uncategorized /by developerPlease help me solve this problem. Thanks. I have attached the image of the question here.
2. Let P2 denote the vector space of all polynomials with real coefficients and of degree at most 2. Define a function T : P2 —> P; by d2 dT(p(w)) — wwflw) + Qfiflw), for all p(m) E P2. In addition, let 5′ = (1, m, .132) be the standard basis of P2. (a) Show that T : P2 —> P2 is a linear operator.(b) Find the matrix A for which [T(p(3:))]3 = A [p(:1:)]3 for all 19(37) E P2.
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Let I I Denote A Pair Of Uncorrelated Zero Mean 2 I Random Vectors Having Cova
/in Uncategorized /by developerLet Y,W denote a pair of uncorrelated, zero-mean(2*1) random vectors having covariance matrix I2
Let I" I`’ denote a pair of uncorrelated , zero – mean | 2* I | random vectors having covariance matrix 1 7 . LetZ = GF + W!where6 = [ 1 1 ]( a ) Determine the LMMISE estimate !" of I’ given I as well as the associated mean – square EllumI bj State the orthogonality principle as it applies in this setting*
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Let P X X 3 X 1 And Q X X 2 X Perform Long Division Of P By Q In The Following F
/in Uncategorized /by developerLet I Be An Ideal Of A Commutative Ring R With Unity If I Is A Prime Ideal Show
/in Uncategorized /by developerLet I be an ideal of a commutative ring R with unity. If I is a prime ideal,show that I[x] is a prime ideal of R[x]. Give an example of a commutative ring with unity and a maximal ideal I of R such that I[x] is not a maximal ideal of R[x].
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