18.06 Problem Set #1

Due Wednesday, Feb.13

1) (a) Find all values of a so that the vectors (a, 4) and (2, 5) are parallel.

(b) Find all values of a so that the vectors (a, 2) and (a, −2) are orthogonal.

(c) Find all values of a so that the vector (1, a, −3, 2) has length 5.

2) Suppose that v and w are non-parallel vectors in ℝ^{2}, with starting point at the origin. Show that the diagonals of the parallelogram determined by v and w bisect each other. (Hint: Use vectors to show that the midpoints of the two diagonals coincide.)

3) Consider the system of equations

−3x + 4y = 8
6x + ty = s

where t ans s are real numbers.

(a) Write the matrix equation for this system.

(b) Find values for s and t so that the system has exactly one solution.

(c) Find values for s and t so that the system has no solutions.

(d) Find values for s and t so that the system has infinitely many solutions.

(e) Give a geometric interpretation of (b), (c), and (d).

4) True or False: If A and B are 2x2 matrices, then (A + B)^{2} = A^{2} + 2AB + B^{2}. If true give a brief explanation, if false give an example where the equality fails.

5) Write down a 3x3 matrix A so that if the vector v = (x, y, z) in ℝ^{3} is multiplied by A, the x and y coordinates of v are unchanged, but the z coordinate becomes zero.