Question

Find the orthogonal complement (using the standard inner-product) of the range (i.e., columnspace) of matrix A

a) Row space of A
b) Null space of A
c) Left null space of A
d) Column space of A

Answer 1

The orthogonal complement of the range of a matrix A is the left null space of A; it consists of all vectors orthogonal to every vector in the column space of A.

Step-by-step explanation:

To find the orthogonal complement of the range (column space) of a matrix A, we need to identify the set of all vectors that are orthogonal to every vector in the range of A. Using the standard inner product (or dot product), we know that if the dot product of two vectors is zero, the vectors are orthogonal. Hence, the answer to the question is c) the left null space of A, which consists of all vectors that when multiplied by A from the left result in the zero vector. This space is orthogonal to the column space of A.

For example, if we have a matrix A and a vector x such that ATx = 0, then x is in the left null space of A. These vectors x, when considered in the context of the matrix equation Ax = b, correspond to the solutions to the homogeneous equation ATx = 0, which are orthogonal to the column space of A.