Exercise 7

Provide a justification for your response to each of the following questions.

  1. Suppose you have \(m\) linearly independent vectors in \(\real^m\text{.}\) Can you guarantee that they form a basis of \(\real^m\text{?}\)

  2. If \(A\) is an invertible \(m\times m\) matrix, do the columns necessarily form a basis of \(\real^m\text{?}\)

  3. Suppose we have an invertible \(m\times m\) matrix \(A\text{,}\) and we perform a sequence of row operations on \(A\) to form a matrix \(B\text{.}\) Can you guarantee that the columns of \(B\) form a basis for \(\real^m\text{?}\)