Consider the matrix P = I - X(XTX)-1X T . If matrix X has 4 rows...

Consider the matrix P = I - X(XTX)-1X T .

If matrix X has 4 rows and 6 columns, what are the dimensions of matrix I?

Prove that P is idempotent.

Expert Solution

Solution:

Rahul Sunny answered 1 year ago

MATLAB Create a matrix named P which has 2000 rows and 100 columns of all zeros....

MATLAB Create a matrix named P which has 2000 rows and 100 columns of all zeros. Replace the zeros in rows 46-58 and columns 3-18 of matrix P from the previous question with ones.

Consider the linear transformation T : P2 ? P2 given by T(p(x)) = p(0) + p(1)...

Consider the linear transformation T : P2 ? P2 given by T(p(x)) = p(0) + p(1) + p 0 (x) + 3x 2p 00(x). Let B be the basis {1, x, x2} for P2. (a) Find the matrix A for T with respect to the basis B. (b) Find the eigenvalues of A, and a basis for R 3 consisting of eigenvectors of A. (c) Find a basis for P2 consisting of eigenvectors for T.

Define and discuss the IPAT relationship (I = P x A x T) as an indicator...

Define and discuss the IPAT relationship (I = P x A x T) as an indicator of humanity’s environmental impact on the planet. Include in your answer some key issues concerning each of the right-hand variables (P, A and T), and the complexities they present for an IPAT assessment.

'(16) Consider the map T : P2(F) →P2(F) deﬁned by T(p(x)) = xp'(x). (a) Prove that...

'(16) Consider the map T : P2(F) →P2(F) deﬁned by T(p(x)) = xp'(x). (a) Prove that T is linear. (b) What is null(T)? (c) Compute the matrix M(T) with respect to the bases B and B' of P2(F), where B = {1,1 + x,1 + x + x2}, B' = {2−x,x + x2,x2}. (d) Use the matrix M(T) to write T(7x2 + 12x−1) as a linear combination of the polynomials in the basis B'.

Find the matrix A' for T relative to the basis B'. T: R3 → R3, T(x,...

Find the matrix A' for T relative to the basis B'. T: R3 → R3, T(x, y, z) = (y − z, x − z, x − y), B' = {(5, 0, −1), (−3, 2, −1), (4, −6, 5)}

Consider the linear transformation T: R^4 to R^3 defined by T(x, y, z, w) = (x...

Consider the linear transformation T: R^4 to R^3 defined by T(x, y, z, w) = (x +2y +z, 2x +2y +3z +w, x +4y +2w) a) Find the dimension and basis for Im T (the image of T) b) Find the dimension and basis for Ker ( the Kernel of T) c) Does the vector v= (2,3,5) belong to Im T? Justify the answer. d) Does the vector v= (12,-3,-6,0) belong to Ker? Justify the answer.

23. Consider the 5 x 6 matrix A whose (i,j)h element is aij i+j; (a) What...

23. Consider the 5 x 6 matrix A whose (i,j)h element is aij i+j; (a) What is A(10:20)? (b) What is k such that A(k) = A(3,4)? (c) Show how A is stored in sparse column format (d) Show how A is stored in sparse row format (e) What is the sparsity ratio?

If X has a t-distribution with df = 14, find P(X < -1.42). NEED STEP BY...

If X has a t-distribution with df = 14, find P(X < -1.42). NEED STEP BY STEP ON HOW TO SOLVE IN EXCEL

Consider the accounting identity: (X – M) = (S – I) + (T – G). Define...

Consider the accounting identity: (X – M) = (S – I) + (T – G). Define each of the individual terms in this identity. What are the implications of this identity for discussions about imbalances in global trade and capital flows?

Consider the initial value problem X′=AX, X(0)=[−4,-2], with A=[−6,0,1,−6] and X=[x(t)y(t)] (a) Find the eigenvalue λ,...

Consider the initial value problem X′=AX, X(0)=[−4,-2], with A=[−6,0,1,−6] and X=[x(t)y(t)] (a) Find the eigenvalue λ, an eigenvector V1, and a generalized eigenvector V2 for the coefficient matrix of this linear system. λ= , V1= ⎡⎣⎢⎢ ⎤⎦⎥⎥ , V2= ⎡⎣⎢⎢ ⎤⎦⎥⎥ $ (b) Find the most general real-valued solution to the linear system of differential equations. Use t as the independent variable in your answers. X(t)=c1 ⎡⎣⎢⎢ ⎤⎦⎥⎥ + c2 ⎡⎣⎢⎢ ⎤⎦⎥⎥ (c) Solve the original initial value problem. x(t)=...

Question