1 Prove that the sum of the modulus squared of the matrix elements of a linear...
1.Derive the matrix formula for the coefficients of a linear model 2.Derive the matrix of velocity exchange for a couple colliding balls 3.Orthogonal operator versus orthogonal matrix (derive the basic property of an orthonormal matrix from a general definition of an orthogonal transformation). Prove existence of a rotation axis for 3D space
MATLAB
Write a code where the elements of a squared matrix size (order) n are equal to the sum of its row and column position (I + j)
2. Work with matrix representations of linear transformations and use knowledge of matrix properties to prove that if a EC is an eigenvalue of a linear operator T:V + V on a (finite-dimensional) inner product space V over C, then ā is an eigenvalue of the adjoint operator T* :V + V. Hint: Check that det (Tij) = det (ij) and utilize this property.
(1) Prove or disprove that if all the elements of a matrix A is
even, the determinant of A is even.
(2) Compute the following determinant
(1) (4 pts) Prove or disprove that if all the elements of a matrix A is even, the determinant of A is even. (2) (2+2 pts) Compute the following determinant (123) (100 A= 1023 B=020 003 co c
4. Let T be the linear operator on F which is represented in the standard ordered basis by the matrix c0 0 01 Let W be the nll space of T - c/. (a) Prove that W is the subspace spanned by 4 (b) Find the monic generators of the ideals S(u;W), S(q;W), s(G;W), 1
4. Let T be the linear operator on F which is represented in the standard ordered basis by the matrix c0 0 01 Let W...
only do (e)-(g) The linear operator L : R3 + R3 is given by its matrix A = Al,s wit respect to the standard basis S = {(1, 22, 23}, where To 0 11 -10- 20 [4 00 (a) Find the characteristic polynomial PL(x) of L; (b) What are the eigenvalues of L and what are their algebraic multiplicities? (e) What are the geometric multiplicities of eigenvalues of L? Is L diagonal- izable? (d) Find a basis B of eigenvectors...
Problem 5 of 7 Consider the standard definitions of sum of squared deviations in the two types of one factor ANOVA discussed in this unit. Prove the following. 1. SST = SSA +SSE (completely randomized ANOVA) 2. SST SSA +SSB SSE (randomized complete block ANOVA) 3. Prove the two alternative formulas for calculating the SSA,SSE,SST in the completely randomized ANOVA. Provide a justification of why someone may prefer to use these formulas against the others that calculate the sum of...
1). Let V be an n-dimensional inner product space, let L be a linear transformation L : V + V. a) Define for inner product space V the phrase "L:V - V" is an orthogonal transforma- tion". b) Define "orthogonal matrix" b) If v1, ..., Vn is an orthonormal basis for V define the matrix of L relative to this basis and prove that it is an orthogonal matrix A.
Problem 5 of 7 Consider the standard definitions of sum of squared deviations in the two types of one factor ANOVA discussed in this unit. Prove the following. 1. SST SSA+SSE (completely randomized ANOVA) 2. SST-SSA+SS SSE (randomized complete block ANOVA) 3. Prove the two alternative formulas for calculating the SSA, SSE, SSr in the completely randomized ANOVA. Provide a justification of why someone may prefer to use these formulas against the others others that calculate the sum of squared...
Linear algebra
6. Prove that if a real matrix An xn satisfies A100-Inxn (the identity matrix), then det(A) ±1.
6. Prove that if a real matrix An xn satisfies A100-Inxn (the identity matrix), then det(A) ±1.