Exercise 3.1.21. If is the plane in described by what is the equation for the plane parallel to through the origin? What is a vector perpendicular to ? Find a matrix for which is the nullspace, and a matrix for which is the row space.

Answer: One way to approach this problem is to find a general solution to the equation and express it as the sum of a homogeneous solution and a particular solution. The corresponding homogeneous system is , which is represented by the matrix

This system has as a basic variable and and as free variables. Setting and we have or . So is one solution to the homogeneous system. Setting and we have or . So is a second solution to the homogeneous system. These vectors are in the nullspace of and serve as a basis for it.

To find the particular solution we set for the general system so that we have or . So is a particular solution to the general system, and the general solution to the equation is then the sum of the particular solution and the homogeneous solution:

The plane is defined by the general solution to . The plane going through the origin corresponds to the homogeneous system and is spanned by the vectors and ; it is the nullspace of the matrix above. The plane is parallel to and is offset from it by the vector .

To find a vector perpendicular to (and ), from the above solution to the homogeneous system we know that the vector (the first and only row of the matrix above) is orthogonal to the vectors and (the basis vectors for the nullspace of ). Therefore the vector is perpendicular to the plane , the nullspace of spanned by and .

Using the vectors and that serve as a basis for the plane we can construct a matrix for which is the row space:

NOTE: This continues a series of posts containing worked out exercises from the (out of print) book Linear Algebra and Its Applications, Third Edition by Gilbert Strang.

If you find these posts useful I encourage you to also check out the more current Linear Algebra and Its Applications, Fourth Edition, Dr Strang’s introductory textbook Introduction to Linear Algebra, Fourth Edition and the accompanying free online course, and Dr Strang’s other books.