Linear Algebra Answers

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.

Are the following vectors linearly independent?

1. (1, 2), (-2, 4)
2. (0, 0, 1), (0, 1, 1), (1, 1, 1)
3. (1, 2), (1, 3), (1, 1)
4. (1, 2, 3), (2, 3, 4)
5. (0, 1, 2), (2, 0, 1), (0, 0, 1), (3, 2, 1)

Use Cayley hamilton theorem to find the values of the matrix

A^8-5A^7+7A^6-3A^5+8A^4-5A^3+8A^2-2A+I