Question

Decide if each of the following statements is true or false. Justify your conclusion with an explanation or ounter-example, as appropriate.

1. If the columns of an m×n matrix are orthonormal, then the linear mapping x ↦A x preserves length.

Answer 1

The statement is true because if the columns of a matrix are orthonormal, the linear mapping x ↦ Ax preserves length.

Step-by-step explanation:

The statement is true.

If the columns of an m×n matrix are orthonormal, it means that each column is a unit vector and every pair of distinct columns are orthogonal to each other.

Let's consider a vector x = [x1, x2, ..., xn] with length ||x||.

When we multiply this vector by the given matrix A, the result Ax will also have length ||Ax||.

Since the columns of the matrix are orthonormal, the dot product of x with each column of A will be 0, except for the column that matches the position of xi.

This means that the length of the resulting vector Ax will be the same as the length of x, which preserves length.

Therefore, the statement is true.