2.75 (a), §2 Solution

Possibilities:

can span 3D space, a plane, a line, or a point.

We will perform Gaussian elimination (row reduction) on the vectors as rows to figure out what space is spanned:

The space spanned is

To answer both parts of the question at once, it is more convenient to perform Gaussian elimination on the system of equations for c₁, c₂, and c₃:

This has the vectors as columns.

Forward elimination:

Back substitution:

From (2), c₂=-c₃; from (1), c₁=-2c₃.

Since c₁, c₂, and c₃ need not be zero, the vectors are linearly dependent.

Taking c₃=-1 gives c₁=2 and c₂=1, hence

09/26/01 0:19:54