a1 and β-: | . | be non-zero vectors, and αΤβ-0. Let aln TL 7L (i)...
Let 4-β 0 0 A=1 0 4-3 024-β where β > 0 is a parameter. (a) Find the eigenvalues of A (note the eigenvalues will be functions of β). (b) Determine the values of β for which the matrix A is positive definite. Determine the values of β for which the matrix A is positive semidefinite. (c) For each eigenvalue of A, find a basis for the corresponding eigenspace. (d) Find an orthonormal basis for R3 consisting of eigenvectors of...
vectors pure and applied. exercise 6.4.2
OIK IIC rather than Example 6.4.1 Let ul, u2 be a basis for F2. The linear map β : F., p given by is non-diagonalisable. hat β is diagonali able with respect to some basis. Then β would have Proof Suppose t matrix representation D=(d, 0 say, with respect to that basis and ß2 would have matrix representation 2 (d2 0 with respect to that basis. However for all xj, so β-0 and β2...
(a) Let P(B1∩B2)>0, and A1∪A2⊂B1∩B2. Then show that P(A1|B1).P(A2|B2)=P(A1|B2).P(A2|B1). (b) Let A and B1 be independent; similarly, let A and B2 be independent. Show that in this case, A and B1∪B2 are independent if and only if A and B1∩B2 are independent. (c) Given P(A) = 0.42,P(B) = 0.25, and P(A∩B) = 0.17, find (i)P(A∪B) ; (ii)P(A∩Bc) ; (iii)P(Ac∩Bc) ; (iv)P(Ac|Bc).
Linear Algebra
Problem # 4 Let A be a 4x4 matrix; the row vectors are a1-(1 230); a2 (452 1):a3-(12 5 0); a4-(2 311) Find a Symmetric matrix S and a skew symmetric T such that A- S+T
Let
A = ( a1 0 ... 0
0 a2 ... 0
... ... ...
0 0 an)
be an n * n matrix, where a1, a2, . . . , an are
nonzero real numbers.
(a) Find the general solution to the system of equations ->
->
x' = A * x
(b) Solve the initial value problem x1(0) =
x2(0) = · · · = xn(0) = k, for some constant
k.
(c) Solve the initial value problem
(x1(0) x2(0)...
= Let cos(6) sin(0) B - sin() cos() and 0 << 27 (i) Calculate the eigenvalues of B. Hence prove that the modulus of the eigenvalues is equal to one. (ii) Calculate the eigenvectors of B.
Let u and v be the vectors shown in the figure to the right, and suppose u and v are eigenvectors of a 2 x2 matrix A that correspond to eigenvalues -2 and 3, respectively. Let T: R2 R2 be the linear transformation given by T(x)-Ax for each x in R2, and let w-u+v. Plot the vectors T(u), T(v), and T(w). 2- u -2 2 4 -2 10- T(v) T(w -10 10 T(u) -10- Ay 10- T(v) T(w) T(u) 10...
(i) Find a non-zero polynomial in Z3 x| which induces a zero function on Z3. f(x), g(x) R have degree n and let co, c1,... , cn be distinct elements in R. Furthermore, let (ii) Let f(c)g(c) for all i = 0,1,2,...n. g(x) Prove that f(x - where r, s E Z, 8 ± 0 and gcd(r, s) =1. Prove that if x is a root of (iii) Let f(x) . an^" E Z[x], then s divides an. aoa1
(i)...
14. Fix a non-zero vector i e R". Let L: R" → R" be the linear mapping defined by L(7) = – 2 proj„(7), for all a E R" (a) Show that if je R", such that j + ð and j ·ñ = 0, then j is an eigenvector of L. What is its eigenvalue? (b) Show that i is an eigenvector of L. What is its eigenvalue? (c) What are the algebraic and geometric multiplicities of all eigenvalues...
(1 point) Let a be a real constant. Consider the equation dx2 dx with boundary conditions y(0)0 and y(2) 0 For certain discrete values of a, this equation can have non-zero solutions. Find the three smallest values of a for which this is the case. Enter your answers in increasing order. a2 , аз Note: You can earn partial credit on this problem
(1 point) Let a be a real constant. Consider the equation dx2 dx with boundary conditions y(0)0...