Question 2 Linear Change-of-Basis For a 3 × 3 change-of-basis matrix, M, used as q(u) = [ u² 1] M qi-1 qi qi+1 what is M for linear interpolation between control points qį and qi+1? Note the order of the powers of u above!
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![Question 2
Linear Change-of-Basis
For a 3 x 3 change-of-basis matrix, M, used as
q(u) = [u² u 1]
qi-1
qi
UB
9i+1,
M
what is M for linear interpolation between control points q; and qi+1? Note the
order of the powers of u above!](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1652026a-6ba6-42d7-9e35-ad3e7516e1ec%2Fe6f328e5-43fc-4df0-85ad-d0c8ec4bac63%2Fbf3xwp_processed.png&w=3840&q=75)
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- "onsdensiks Consider the m x n-matrix A and the vector be Rm that are given by A = [aij], 6 = [bi] where aij = (–1)++i(i – j), b; = (-1)' for i = 1,... m and j = 1,..· , n. %3D Note that aij is the entry of A at the i-th row and the j-th column. Consider the following condition on vectors iE R": (Condition1) Aa õ. Write and run a python Jupyter notebook using the PULP package, to check whether there is a vector iE R" that satisfies (Condition1), • when m =n = 10, and if it exists, find one. Submit your screenshots or the pdf file of your Jupyter Notebook; it should show both the codes and the results. [For credits, you must write a python Jupyter notebook to solve this problem.] Python Hint: Start with giving М-10 N=10 You may want to use Python lists combined with 'forloop'. For example, column=[j+1 for j in range(N)] row=[i+1 for i in range(M)] Then, one can define the matrix A and the vector 6, using Python dictionaries combined with 'forloop' as a= {i:{j:(-1)**(i+j)*(i-j) for j in…Write a code to perform an LU decomposition of the coefficient matrix [A] (given below) using L-U decomposition with Gauss elimination. Your code should output [L] and [U] as well as verify that: Use the following MATLAB matrix function only to check your answer: where A = [1 3 6; 4 5 8; 1 2 3].. The determinant of an n X n matrix can be used in solving systems of linear equations, as well as for other purposes. The determinant of A can be defined in terms of minors and cofactors. The minor of element aj is the determinant of the (n – 1) X (n – 1) matrix obtained from A by crossing out the elements in row i and column j; denote this minor by Mj. The cofactor of element aj, denoted by Cj. is defined by Cy = (-1y**Mg The determinant of A is computed by multiplying all the elements in some fixed row of A by their respective cofactors and summing the results. For example, if the first row is used, then the determi- nant of A is given by Σ (α(CI) k=1 Write a program that, when given n and the entries in an n Xn array A as input, computes the deter- minant of A. Use a recursive algorithm.
- If there is a non-singular matrix P such as P-1AP=D, matrix A is called a diagonalizable matrix. A, n x n square matrix is diagonalizable if and only if matrix A has n linearly independent eigenvectors. In this case, the diagonal elements of the diagonal matrix D are the eigenvalues of the matrix A. A=({{1, -1, -1}, {1, 3, 1}, {-3, 1, -1}}) : 1 -1 -1 1 3 1 -3 1 -1 a)Write a program that calculates the eigenvalues and eigenvectors of matrix A using NumPy. b)Write the program that determines whether the D matrix is diagonal by calculating the D matrix, using NumPy. #UsePythonThe matrix [A] has a norm of ||A||=3.2 x 10° and the norm of the inverse of that matrix is ||A-4||=2.5 x 1012 If a solution is obtained for a system of equations [A]{x}={b} using this same matrix [A] and double precision computations, then O the solution is very sensitive to input error O the solution is insensitive to input error O small residuals indicate accurate answers O small residuals do not indicate accurate answers O the system is ill-conditionedProve that f(n)= {floor function of sqrt(n)} - { floor function of sqrt(n-1)} is a multiplicative function, but it is not completely multiplicative.
- Let f ∈ C+ 2π with a zero of order 2p at z. Let r>p and m = n/r. Then there exists a constant c > 0 independent of n such that for all nsufficiently large, all eigenvalues of the preconditioned matrix C−1 n (Km,2r ∗ f)Tn(f) are larger than c.Use a graphing utility or computer software program with matrix capabilities to write v as a linear combination of u1, u2, u3, u4, u5 and u6. Then verify your solution.Verify that the full perspective to canonical matrix Mprojection takes (r, t, n) to (1, 1, 1).
- Do all mxn matrix has a determinant?A What is the product of matrices M=(1,0)(0,-3) and N=(0,1)(0,-3)? B What is the difference of matrices M=(1,0,2)(0,-3,1) and N=(0,1,2)(0,0,-3)? C What is the a(3) term of the sequence {a(n)} where a(n) = 2n - 6? 1.D What is the range of a function that assigns to a 16-bit bit string the number of one bits in the bit string? E What is the sum of matrices M=(1,0,2)(0,-3,1) and N=(0,1,2)(0,0,-3)? F If f(x) = 2x + 3, find f(S) if S={0,2,5}. G What is the transpose of matrix M=(1,0,2)(0,-3,1)?Use K-map to obtain the minimized product of sums form of the function f(a,b,c,d) = SEGMA (Q. 1. 2. 3. 4. 6. 8. 9. 10. 11. 3, 15). Use the editor to format your answer