Problem 1

Let

(a). How Many Solutions Are There?

The system has infinitely many solutions

(b). A Vector is Nonnegative if All of Its Coordinates Are Nonnegative, and Nonpositive if All of Its Coordinates Are Nonpositive. Describe All the Nonnegative Solutions, or Explain why there Are no Nonnegative Solutions

There is only one nonnegative solution .

(c). Describe All the Nonpositive Solutions, or Explain why there Are no Nonpositive Solutions

There are no nonpositive solutions as can not be greater than 2 and simultaneously less than 0.

(d). Find a Linear Equation That, if Added to the given System of Four Linear Equations, Would Form a Consistent System with Exactly One Solution

Problem 2

(a). Let Be a Matrix. how Many Solutions Might there Be to the Matrix Equation

There are either 0 solutions (if the equation is inconsistent where one row reduces to for some constant ) or infinitely many solutions as there would be a free variable. There can not be a unique solution.

(b). Solve the following

Let , , , , , , and .

Let have row echelon form and satisfy , , and . Find all solutions to , or explain where there are no solutions.

Problem 3

(a). Let Be the line in . Are there Any Lines Parallel to through the point ? Express Them in Parametric Form, or Explain why there Are no such Lines

All lines parallel to are of the form , plugging in the point we get a solution . Expressing this in parametric form with we get the solution , .

(b). Let Be as in part (a). Are there Any Lines Perpendicular to L through the point ? Express Them in Parametric Form, or Explain why there Are no such Lines (Here and in part (d), You May Take Perpendicular to Mean "at Right angles".)

All lines perpendicular to are of the form , plugging in the point we get a solution . Expressing this in parametric form with we get the solution ,

(c). Let Be the Surface in . Are there Any Lines Parallel to through the point ? Express Them in Parametric Form, or Explain why there Are no such Lines

All the parallel lines to that pass through the point fall within the plane for some constant , plugging in the point we get , therefore all parallel lines to that pass through the point fall on the plane .

Treating and we can parameterize the plane by first writing in terms of as ,

where is some "slope" constant which varies by solution.

(d). Let Be the Surface as in part (c). Are there Any Lines Perpendicular to through the point ? Express Them in Parametric Form, or Explain why there Are no such Lines

Again all perpendicular lines to that pass through the point must be parallel to the normal vector of the plane . We therefore know that we can parameterize the solutions as

This only gives us a single line as there is no unknown parameter.