Linear Algebra Answers

Let T: R^3 to R^3 be the linear transformation defined by

T(x,y,z)= (-x,x-y, 3x+2y+z)

Check whether T satifies the polynomial (x-1)(x+1)^2. Also the find of minimal polynomial of T.

Check that {1,(x+1),(x+1)^2} is a basis of the vector space of polynomial over R of degree at most 2. Find the coordinate of 3+x+2x^2 with respect to the basis.

Let P = [ -1 4 5] . Determine P^-1 using

[ 0 2 -3]

[ 0 0 8]

Cayley- Hamilton theorem. Further use P^-1 to express (x1, x2, x3) in terms of (-1,0,0), (4,2,0), ( 5,-3,8)

Find an orthonormal basis of R^3, of which (0,3√13, 2√13) is one element

For each of the following functions determine the inverse image of T = {x ∈ R : 0 ≤ x 2 − 25}. 

1. f : R → R defined by f(x) = 3x³.

2. g : R + → R defined by g(x) = ln(x).

3. h : R → R defined by h(x) = x − 9.

1.      Use Gaussian elimination to solve the system of linear equations

300x₁ 112x₂ 109x₃ = 521

252x₁ 156x₂ 330x₃ =738

108x₁ -123x₂ 121x₃ =106

2.     Solve the following system linear equations by Gauss Jordan Method

x +y +z = 5

2x +3y +5z = 8

4x + 5z = 2

the upper triangular n x n matrices with no zeros on the diagonal

5x +2y +z =-8

x -2y -3z =0

-x +y +2z =3

Solved this problem by using Gauess Gordan method.

Find 2×2 matrix A that maps (1,3)^T and (1,4)^T into (-2,5)^T and (3,-1)^T, respectively

Known Matrix:

1. Determine the characteristic equation det(A − λI) = 0 of the matrix above
2. Determine the eigenvalues of the matrix and the basis of the eigenspace

Note: In matrix B, let the value of a be so that the eigenvalues and the basis of the eigenspace are dependent on a.