March 21, 2003
(1) (9 points) This problem concerns the vectors , , .
(b) (1)(2) + (3)(1) + (2)(3) = 11
(c) Two of the vectors u, v, and w are orthogonal. Which two, and why?
v and w because their dot product is zero.
(2) (10 points) Find the distance between the point in and the plane .
Call the above point q =. A point on the plane is p = . A normal vector to the plane is n = . The vector pointing from p to q is r = q - p = . The distance from q to the plane is .
(3) (10 points) Find the solution of the following linear system. Write the solutions in vector form.
(a) (10 points) Consider the set of vectors in . Is this a line or a plane? If it is a line, write its equation in vector from. If it is a plane, write its equation in normal form.
Notice the second vector is a multiple of the first so it can be eliminated. Then = is line through the origin. It's vector form is .
(b) (10 points) Consider the set of vectors in . Is this a line or a plane? If it is a line, write its equation in vector from. If it is a plane, write its equation in normal form.
This time the two vectors are not multiples of each other, so we are dealing with a plane throught the origin.
Its normal form must be ax + by + cz = 0, so we just need to find a b and c. Since the two vectors are on the plane we get the following system.
The solutions are thus a = -c and b = c, with c free. Set c = 1 to get b = 1 and a = -1.
The general form of the plane is thus -x + y + z = 0.
Its normal form is thus
(5) (10 points) Suppose A, B and X are invertible n-by-n matrices. Solve the following equation for X.
(6) This problem concerns the vectors , , .
(a) (10 points) Are these vectors linearly independent or linearly dependent? Show your work.
Setting this up in the ususal way, we look at the solutions of the system
Setting up the augmented matrix,
So you see there's only the trivial solution, so the vectors are linearly independent.
(b) (5 points) Is the span of these vectors equal to ? Why or why not?
NO. Notice the third component of each vector is 0. Any linear combination of them will thus have a zero in the third component. Since o has vectors whose third components are not zero, the three vectors in this problem cannot span it.
(7) (10 points) Suppose A is a 3-by-4 matrix. Are the columns of A linearly independent, linearly dependent, or is there not enough information to say? Explain.
Let A = [a b c d] where a, b, c and d are vectors in . In testing for independence, we look at the solutions of the system xa + yb + zc +wd = 0. This is a homogeneous system of three equations in 4 variables. Solving it will thus result in free variables, so there will be a nontrivial solution. Thus the columns are linearly dependent.
(8) (10 points) This problem concerns the invertible matrix
(a) Find .
(b) (6 points) Use your answer to part (a) above to find a solution to the equation .