Question

please choose your favorite, unique plane in R³ that passes though the origin (0,0,0). (An example plane is: x + 5y – z = 0. You can use your Module Four Discussion Forum plane through the origin if desired.)

1. What is the equation for your plane? (4 points)
2. What is a basis for the subspace of R³ formed by your plane? Hint: (y and z are free variables.) (8 points)
3. Identify three non-zero vectors on the plane. Do not choose a vector that is a scale multiple of another vector that you have chosen. For example, if (1,3,7) is one vector on your plane, do not choose (5,15,35) or any other multiple of (1,3,7). (Make sure that each vector satisfies the equation for your plane.) What are the three vectors that you have selected? (6 points)
4. Create a 3x3 matrix A where each of your selected vectors on the plane is a column in the matrix. (8 points for entire problem)
   • What is your matrix A? Calculate |A| to verify that |A| = 0.
5. Determine a basis for the column space of A (i.e. for CS(A)). Explain your approach. (6 points)

(Suggestion: Typically, the approach of finding a basis for the row space (RS(A) of A^T results in a simpler basis (i.e. more zeros), which will be useful in a later problem in this exam.)

Note that the CS(A) is the same subspace of R³ as the set of vectors in your plane.

Answer 1