Algebra Problems

1. Do any two of a - c. Consider the lines given by the equations x − 2y = 2 and 2x + y = −2. a. Find the angle between the lines. b. Find vector-parametric equations for each of the lines. c. Find the point where the lines intersect. 2. Do any two of a - c. Let u = 1 1 1 1 , v = 1 0 1 1 , andw = 1 1 0 1 a. Find the angle θ between u and v. b. Solve the equation 2u + 4w + 5x = −3v for x. c. Find the components of w that are, respectively, parallel to and perpendicular to v. 3. Consider the following system of linear equations 2x + y + z = 1 x + y = 1 x + 2y − z = 1 a. Find all solutions, if any, of this system. b. Determine whether the vectors 2 1 1 1 1 2 , and 1 0 −1 are linearly dependent. 4. As in 2, let u = 1 1 1 1 , v = 1 0 1 1 , andw = 1 1 0 1 . In addition, let x = 1 . a. Determine whether x ∈ Span{u, v, w}. b. Determine whether u, v and w are linearly dependent or independent.