Exercise 7

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

1. Suppose you have $m$ linearly independent vectors in $\real^m$. Can you guarantee that they form a basis of $\real^m$?

2. If $A$ is an invertible $m\times m$ matrix, do the columns necessarily form a basis of $\real^m$?

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

4. Suppose you have a set of 10 vectors in $\real^{10}$ and that every vector in $\real^{10}$ can be written as a linear combination of these vectors. Can you guarantee that this set of vectors is a basis for $\real^{10}$?