Went ahead and did the HW the teach recommended but she did not post the answers and I would like to see if im on the right track.
Problem 1: Are the vectors (2,−1,−3), (3, 0,−2), (1, 1,−4) linearly independent?
Problem 2 : Is the set {x E R^4 : (x1 − x2)^2 + x3^4 = 0} a subspace? Justify
Problem 3 : Solve the system
2x1 + x2 + 3x3 + x4 = 1
x2 + 4x3 + x4 = 1
3x1 + 2x4 = 1.
Problem 4 : Find an othonormal basis of the subspace S = {x E R^4 :
x1 + x2 + x3 + x4 = x1 − x2 − 2x3 + x4 = x2 + 3x3 = 0}. What is the dimension on S?
Problem 5 : Let
A = 0 2 −1
1 2 3
2 2 −1
Compute A−1.
Problem 6: Find the eigenvalues and eigenvectors of
A = −1 1
1 −2
Problem 7: Find the eigenvalues and eigenvectors of
A = −2 1
−4 −2
To check if you are on the right track, you need to understand the concepts and techniques required for each problem. Here's a step-by-step breakdown of how you can approach each problem:
Problem 1: To determine if the vectors (2, -1, -3), (3, 0, -2), and (1, 1, -4) are linearly independent, you can create a matrix with these vectors as columns and check for linear dependence by finding its determinant. If the determinant is non-zero, the vectors are linearly independent.
Problem 2: To determine if the set {x E R^4 : (x1 − x2)^2 + x3^4 = 0} is a subspace, you need to verify if it satisfies the three conditions of a subspace: closure under addition, closure under scalar multiplication, and contains the zero vector. You can analyze these conditions by substituting values into the equation and checking if they satisfy the conditions.
Problem 3: To solve the system of equations, you can represent the system as an augmented matrix and perform row operations to transform it into row-echelon form or reduced row-echelon form. This will help you determine the solution to the system, whether it has a unique solution, no solution, or infinitely many solutions.
Problem 4: To find an orthonormal basis of the subspace S and its dimension, you can start by finding a basis for the subspace S. Then, using the Gram-Schmidt process, you can orthogonalize the basis vectors by making them mutually orthogonal. Finally, you can normalize the vectors to obtain an orthonormal basis. The dimension of the subspace S will be equal to the number of linearly independent vectors in the basis.
Problem 5: To compute the inverse of a matrix A, you can use various techniques such as row reduction, cofactor expansion, or using the adjugate matrix. The method will depend on the size and properties of the matrix A.
Problem 6: To find the eigenvalues and eigenvectors of a matrix A, you need to solve the characteristic equation by finding the determinant of (A - λI) (where I is the identity matrix and λ is the eigenvalue) and setting it equal to zero. Once you find the eigenvalues, you can substitute them back into (A - λI) to find the eigenvectors.
Problem 7: Similarly to Problem 6, to find the eigenvalues and eigenvectors of matrix A, you need to find the determinant of (A - λI) and solve for λ. Once you have the eigenvalues, you can substitute them back into (A - λI) to find the eigenvectors.
Remember, these steps provide a general approach to each problem. Depending on your specific class or textbook, there may be additional techniques or methods that could be used to solve these problems.