##### Exercise7

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 $$\real^m\text{?}$$

in-context