Let L: IRn IRm be a linear mapping. What can we say about the injectivity, the surjectivity, & the bijectivity of L if:
a) n > m
b) n = m
c) n < m
Also, give a simple example for each case.
Let 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.
P = 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.
Please help me solve those questions, the questions are shown in the attachments (6, 7, 8)
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
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.- 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
Can anyone help me with this question please? The question 3 a) in assignment 6 is in the picture below.
Thank you so much!
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 .
Please 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.
Let 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*
Let p(x) = x 3 + x + 1, and q(x) = x 2 + x. Perform long division of p, by q, in the following fields. (I.e. write p(x) = f(x)q(x) + r(x) where degr < degq.) (a) F = R (b) F = Z3 (Hint: Use (a).) (c) F = Z5 (Hint: Use (a).)
Let 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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